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Preface to the Series

Contributions to Mathematical and Computational Sciences

Mathematical theories and methods and effective computational algorithms are crucial in coping with thechallenges arising in the sciences and in many areas of their application. New concepts and approachesare necessary in order to overcome the complexity barriers particularly created by nonlinearity, high-dimensionality, multiple scales and uncertainty. Combining advanced mathematical and computationalmethods and computer technology is an essential key to achieving progress, often even in purely theoreticalresearch.

The term mathematical sciences refers to mathematics and its genuine sub-fields, as well as to scientificdisciplines that are based on mathematical concepts and methods, including sub-fields of the natural andlife sciences, the engineering and social sciences and recently also of the humanities. It is a major aim ofthis series to integrate the different sub-fields within mathematics and the computational sciences, and tobuild bridges to all academic disciplines, to industry and other fields of society, where mathematical andcomputational methods are necessary tools for progress. Fundamental and application-oriented researchwill be covered in proper balance.

The series will further offer contributions on areas at the frontier of research, providing both detailedinformation on topical research, as well as surveys of the state-of-the-art in a manner not usually possiblein standard journal publications. Its volumes are intended to cover themes involving more than just a single“spectral line” of the rich spectrum of mathematical and computational research.

The Mathematics Center Heidelberg (MATCH) and the Interdisciplinary Center for Scientific Computing(IWR) with its Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences(HGS) are in charge of providing and preparing the material for publication. A substantial part of the mate-rial will be acquired in workshops and symposia organized by these institutions in topical areas of research.The resulting volumes should be more than just proceedings collecting papers submitted in advance. Theexchange of information and the discussions during the meetings should also have a substantial influenceon the contributions.

This series is a venture posing challenges to all partners involved. A unique style attracting a largeraudience beyond the group of experts in the subject areas of specific volumes will have to be developed.

Springer Verlag deserves our special appreciation for its most efficient support in structuring and initiat-ing this series.

Heidelberg University, Germany Hans Georg BockWilli Jäger

Otmar Venjakob

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Preface

Iwasawa Theory is one of the most active fields of research in modern Number Theory. The great interestin Iwasawa Theory is reflected by the very successful bi-annual series of international conferences, start-ing in 2004 in Besancon and continuing in Limoges, Irsee and Toronto with scientific committee formedby John Coates, Ralph Greenberg, Cornelius Greither, Masato Kurihara, and Thong Nguyen Quang Do.The Iwasawa Conference 2012, organized by Otmar Venjakob and Thanasis Bouganis, took place in Hei-delberg (July 30-August 3), with more than 120 participants. It was supported by the Mathematics CenterHeidelberg (MATCH) and by the European Research Council (ERC) Starting Grant IWASAWA of OtmarVenjakob. This volume, Iwasawa Theory 2012- State of the Art and Recent Advances, contains research andoverview articles contributed by speakers and participants of the conference as well as lecture notes froman introductory mini-course given by Chris Wultrich and Xin Wan, which took place the week before theconference.

One can argue that Iwasawa Theory has its roots in the early 19th century and in the work of ErnstKummer (29 January 1810-14 May 1893), who studied the class number of the cyclotomic field Q(ζp),in his approach to prove Fermat’s Last Theorem. Kummer not only provided a solution to Fermat’s LastTheorem for a large class of prime exponents, but also a link between the p-divisibility of the class numberof Q(ζp) and the values of the Riemann zeta function at the negative integers. This link between arithmeticexpressions and special values of zeta functions, which was later refined by the work of Herbrand and Ribet,lies in the heart of modern number theory. It is the earliest example of an array of deep relations betweenarithmetic expressions and L-values, highly conjectural, the most celebrated of which is the Conjecture ofBirch and Swinnerton-Dyer.

However it was Kenkichi Iwasawa (September 11, 1917- October 26, 1998), and his Main Conjecture,which completely transformed our view of the arithmetic of cyclotomic fields. Indeed Iwasawa, inspired bythe work of Andre Weil on the Zeta Function of varieties over finite fields, initiated the systematic investi-gation of the p-part of the class number in the cyclotomic extension of Q. Not only he managed to provehis deep theorems with respect to the growth of the p-part of the class number in such extensions but alsoformulated his Main Conjecture, which relates the size of particular Galois module to the Kubota-Leopoldp-adic L-function. This conjecture formed the prototype for an array of Main Conjectures, which predict adeep relation between p-adic L functions and arithmetic invariants of abelian varieties or, even more gener-ally, motives.

The Main Conjecture for cyclotomic fields is now a theorem and there has been considerable progressin other fronts, as for example the Main Conjectures for CM fields, for elliptic modular forms and the MainConjectures for abelian varieties over function fields. The proofs of all these Main Conjectures involve animpressive combination of various strands of pure mathematics such as K-theory, automorphic forms, alge-braic geometry, contributing enormously to the popularity of the subject. Iwasawa Theory has not stoppedto grow in complexity and generalization. Undoubtedly the work of Hida, and his investigation of whatnowadays is called Hida-families, has transformed the way that we view Iwasawa theory today. There hasbeen also great interest in extending Iwasawa Theory to a non-abelian setting, where one is interested inthe arithmetic behaviour of the underlying motive over a p-adic Lie extension. A vast generalization of theMain Conjectures to this non-abelian setting has now been formulated and there have been already some

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first results both in the number field and in the function field case.

It is exactly these astonishing new and rapid developments that the Iwasawa Conference series tries toaddress. The main aim is to bring together experts from different strands in and closely related to Iwasawatheory to report on recent developments and exchange ideas. The Iwasawa Conference series has also estab-lished a tradition of very lively and pleasant meetings, and this tradition was strengthened by the Iwasawa2012 conference. Events such as the half day long cruise on the river Neckar undoubtetly helped create afriendly and stimulating atmosphere among the participants of the conference.

The week before the Iwasawa 2012 Conference a preparatory mini-course took place, given by ChrisWultrich and Xin Wan, aimed to graduate students and newcomers to the field. Chris Wultrich offered anoverview of some basic aspects of Iwasawa Theory and Xin Wan an introduction to the work of Skinner andUrban on the Main Conjecture for elliptic modular forms. Their lecture notes, Overview of some IwasawaTheory by Chris Wultrich and Introduction to Skinner-Urban’s Work on the Main Conjecture for GL2 byXin Wan, appear now in this volume. The organizers would like to take the opportunity in this preface tothank them again for their excellent lecture series in the summer of 2012 and their contributions to thisvolume.

The talks given in the Iwasawa 2012 conference covered the wide range of development in IwasawaTheory in the recent years, and it was complemented by a poster session. Not every contribution in thisvolume is related to some talk given during the conference. Some of the contributions are survey articles,some other are original research articles appearing for first time in printed form.

Acknowledgements: It is of course the effort and work of the contributors that make this volume possi-ble. The editors are grateful to them. Moreover the editors would like to thank the referees for their work.Taking the opportunity of this volume the editors thank once again the speakers and the participants ofthe Iwasawa 2012 conference and the preparatory lecture series. Further the editors would like to expresstheir gratitude to Birgit Schmoetten-Jonas since her help for organizing the Iwasawa 2012 conference andediting this volume has been, nothing less than, indispensable. Finally it is a pleasure to thank MATCH andERC for the financial support as well as Mrs Allewelt and Dr Peters from Springer Verlag for the excellentcollaboration while editing this volume.

The Editors,

Otmar Venjakob and Thanasis Bouganis

Contents

Overview of some Iwasawa theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Christian Wuthrich

1 Iwasawa theory of the class group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1 Zp-extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2 The Iwasawa algebra and its modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3 Proof of Iwasawa’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4 Stickelberger elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5 p-adic L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.6 The main conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.7 Cyclotomic units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.8 Vandiver’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Iwasawa theory for elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Selmer groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Iwasawa theory for the Selmer group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4 Mazur-Stickelberger elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5 The p-adic L-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.6 The main conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 The leading term formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1 Control theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 p-adic heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 Leading term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Selmer groups for general Galois representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.1 Local conditions at places away from p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Local conditions at places above p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3 The Selmer groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Kato’s Euler system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.1 Construction of the Euler system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2 Relation to p-adic L-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.3 Euler system method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Introduction to Skinner-Urban’s Work on the Iwasawa Main Conjecture for GL2 . . . . . . . . . . . . 35Xin Wan

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 Main conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.1 Families of Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.2 Characteristic Ideals and Fitting ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3 Selmer Groups for Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.4 p-adic L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.5 The Main Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

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2.6 Two and Three-Variable Main Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.7 Comment on the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.8 A Theorem of Ribet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Hermitian Modular Forms on GU(n,n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1 Hermitian Half Space and Automorphic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Hida Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Eisenstein Series on GU(2,2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.1 Klingen Eisenstein Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 p-adic Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Galois Representations and Lattice construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.1 Galois Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2 Galois Argument: Lattice Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6 Doubling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.1 Siegel Eisenstein Series on GU(n,n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.2 Some Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.3 Pullback Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7 Constant Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 p-adic Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 Fourier-Jacobi Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

9.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509.2 Backgrounds for Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519.3 Coprime to the p-adic L-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

10 Generalizations of the Skinner-Urban Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

On extra zeros of p-adic L-functions: the crystalline case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Denis Benois

0.1 Extra zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550.2 Extra zero conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560.3 Selmer complexes and Perrin-Riou’s theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580.4 The plan of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611.1 (ϕ,Γ )-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611.2 Crystalline representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2 The exponential map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.1 The Bloch–Kato exponential map ([4], [22], [31]) . . . . . . . . . . . . . . . . . . . . . . . . . . 652.2 The large exponential map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3 The L -invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.1 Definition of the L -invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.2 L -invariant and the large exponential map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4 Special values of p-adic L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.1 The Bloch–Kato conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2 p-adic L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5 The module of p-adic L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.1 The Selmer complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.2 The module of p-adic L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

On Special L-Values attached to Siegel Modular Forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Thanasis Bouganis

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032 Siegel Modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

2.1 Integral weight Siegel modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1042.2 Half-integral weight Siegel modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3 Theta and Eisenstein Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Contents 7

3.1 Theta series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063.2 Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063.3 Eisenstein series of integral weight. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103.4 Eisenstein series of half integral weight. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4 The L-function attached to a Siegel Modular Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165 The Rankin-Selberg Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176 Petersson Inner Products and Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197 Algebraicity results for Siegel Modular Forms over totally real fields . . . . . . . . . . . . . . . . . . 1268 Some Remarks on the Doubling Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Modular symbols in Iwasawa theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133Takako Fukaya, Kazuya Kato, and Romyar Sharifi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1332 The case of Gl2 over Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

2.1 From modular symbols to cup products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1352.2 Passing up the modular and cyclotomic towers: the map ϖ . . . . . . . . . . . . . . . . . . . 1372.3 Zeta elements: ϖ is “Eisenstein” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1392.4 Ordinary homology groups of modular curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1392.5 Refining the method of Ribet and Mazur-Wiles: the map ϒ . . . . . . . . . . . . . . . . . . . 1412.6 The conjecture: ϖ and ϒ are inverse maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1442.7 The proof that ξ ′ϒ ϖ = ξ ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

3 The case of Gl2 over Fq(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1473.1 From modular symbols to cup products: the map ϖ . . . . . . . . . . . . . . . . . . . . . . . . . 1483.2 Working with fixed level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493.3 Zeta elements: ϖ is “Eisenstein” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1503.4 Homology of Drinfeld modular curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1503.5 The map ϒ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1513.6 The conjecture: ϖ and ϒ are inverse maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1533.7 The proof that ξ ′ϒ ϖ = ξ ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

4 What happens for Gld? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1544.1 The space of modular symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1544.2 Questions for the general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Weber’s Class Number One Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161Takashi Fukuda, Keiichi Komatsu and Takayuki Morisawa

1 Weber’s class number one problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1612 Composites of Zp-extensions of Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

On p-adic Artin L-functions II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165Ralph Greenberg

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1652 Basic results concerning the Selmer group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1663 The definition of p-adic Artin L-functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1724 Relationship of Selmer groups to p-adic L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Iwasawa µ-invariants of p-adic Hecke L-functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179Ming-Lun Hsieh

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1791.1 p-adic Hecke L-function for CM fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1791.2 The µ-invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1801.3 p-adic L-functions and Selmer groups for CM elliptic curves . . . . . . . . . . . . . . . . . 181

2 Ferrero-Washington theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1833 Hida’s theorem on the µ-invariants of anticyclotomic p-adic Hecke L-functions . . . . . . . . . 184

8 Contents

3.1 Linear independence for p-adic modular forms modulo p . . . . . . . . . . . . . . . . . . . . 1853.2 Period integrals and Fourier coefficients of toric forms . . . . . . . . . . . . . . . . . . . . . . 186

4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1884.1 The cyclotomic derivative of Lχ,Σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1884.2 Anticyclotomic p-adic Rankin-Selberg L-functions . . . . . . . . . . . . . . . . . . . . . . . . . 189

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

The p-adic height pairing on abelian varieties at non-ordinary primes . . . . . . . . . . . . . . . . . . . . . 191Shinichi Kobayashi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1912 Norm subgroup of formal groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1923 Norm construction of the p-adic height pairings on abelian varieties at non-ordinary primes196

3.1 Submodules corresponding to splittings of the Hodge filtration . . . . . . . . . . . . . . . 1963.2 The Zarhin-Nekovár construction of p-adic height pairings . . . . . . . . . . . . . . . . . . . 2013.3 Norm construction of p-adic height pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

4 An application for the p-adic Gross-Zagier formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

Iwasawa modules arising from deformation spaces of p-divisible formal group laws . . . . . . . . . . . 209Jan Kohlhaase

0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2091 Formal group laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2102 Deformation problems and Iwasawa modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2133 Rigidification and local analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2174 Non-commutative divided power envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

The structure of Selmer groups of elliptic curves and modular symbols . . . . . . . . . . . . . . . . . . . . . 227Masato Kurihara

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2271.1 Structure theorem of Selmer groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2271.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

2 Selmer groups and p-adic L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2312.1 Modular symbols and p-adic L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2312.2 Selmer groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2322.3 An analogue of Stickelberger’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2352.4 Higher Fitting ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

3 Review of Kolyvagin systems of Gauss sum type for elliptic curves . . . . . . . . . . . . . . . . . . . 2373.1 Some definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2373.2 Euler systems of Gauss sum type for elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . 2393.3 Kolyvagin derivatives of Gauss sum type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2393.4 Construction of Kolyvagin systems of Gauss sum type . . . . . . . . . . . . . . . . . . . . . . 241

4 Relations of Selmer groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2424.1 Injectivity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2424.2 Relation matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

5 Modified Kolyvagin systems and numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2445.1 Modified Kolyvagin systems of Gauss sum type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2445.2 Proof of Theorem 1.2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2455.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2485.4 A Remark on ideal class groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Contents 9

p-adic integration on ray class groups and non-ordinary p-adic L-functions . . . . . . . . . . . . . . . . . . 253David Loeffler

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2532 Integration on p-adic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

2.1 One-variable theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2542.2 The case of several variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2552.3 Groups with a quasi-factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

3 Distributions on ray class groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2573.1 Criteria for distributions of order r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2583.2 A converse result for totally real fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2593.3 Imaginary quadratic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2613.4 Fields of higher degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2613.5 A representation-theoretic perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

4 Construction of p-adic L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2634.1 Mazur–Tate elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2634.2 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2644.3 Consequences for p-adic L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

5 The ap = 0 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

On equivariant characteristic ideals of real classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269Thong Nguyen Quang Do

0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2691 Generators and relations, and Fitting ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

1.1 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2711.2 A kummerian description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

2 Semi-simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2733 Equivariant study of the non semi-simple case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

3.1 Perfect torsion complexes and characteristic ideals . . . . . . . . . . . . . . . . . . . . . . . . . . 2753.2 Iwasawa cohomology complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2753.3 The limit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2763.4 The determinant of H2

Iw(F∞,Zp(m)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2773.5 The Fitting ideal of H2

Iw(F∞,Zp(m)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Nearly overconvergent modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283Eric Urban

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2832 Nearly holomorphic modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

2.1 Classical definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2842.2 Sheaf theoretic definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2862.3 Rational and integral structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2872.4 Differential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2892.5 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2912.6 Rationality and CM-points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

3 Nearly overconvergent forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2923.1 Katz p-adic modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2923.2 Nearly overconvergent forms as p-adic modular forms . . . . . . . . . . . . . . . . . . . . . . 2933.3 Families of nearly overconvergent forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

4 Application to Rankin-Selberg p-adic L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3024.1 Review on Rankin-Selberg L-function for elliptic modular forms . . . . . . . . . . . . . . 3024.2 The p-adic Petersson inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3034.3 The nearly overconvergent Eisenstein family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3044.4 Construction of the Rankin-Selberg L-function on E×E×X . . . . . . . . . . . . . . . . . 304

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

10 Contents

Noncommutative L-functions for varieties over finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311Malte Witte

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112 Waldhausen Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3133 The K-Theory of Adic Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3144 Perfect Complexes of Adic Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3175 On SK1(Z`[G][[T ]]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3206 L-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3237 The Grothendieck trace formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

On Z-zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331Zdzisław Wojtkowiak

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312 Formulation of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3323 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3344 Relations with Iwasawa theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

Overview of some Iwasawa theory

Christian Wuthrich

Introduction

These are the notes to lectures given at Heidelberg in July 2012. The intention was to give an conciseoverview of some topics in Iwasawa theory to prepare the participants for the conference. As a consequence,they will contain a lot of definitions and results, but hardly any proofs and details. Especially I would liketo emphasise that the word “proof” should be replaced by “sketch of proof” in all cases below. Also, I haveno claim at making this a complete introduction to the subject, nor is the list of references at the end. Forthis the reader might find [Gre01] a better source.

The talks were given in four sessions, which form the four sections of these notes. We start by theclassical Iwasawa theory for the class group, including the fundamental result of Iwasawa on the growthof class groups in Zp-extensions. We also describe Stickelberger elements, cyclotomic units and the mainconjecture. This first section also contains the basic facts about Iwasawa algebras.

The second section introduces Iwasawa theory for elliptic curves by studying the growth of the Selmergroup. We define Mazur-Stickelberger elements and the p-adic L-functions and state the main conjecture inthis context. The third section includes the proof of the control theorem for Selmer groups (in the ordinarycase) and the formula for the leading term of the characteristic series of the Selmer group. The last sectionshows how one generalises Selmer groups to various Galois representations. We conclude with a rough andshort explanation about Kato’s Euler system.

Acknowledgements It is my pleasure to thank Thanasis Bouganis, Sylvia Guibert, Chern-Yang Lee, Birgit Schmoetten-Jonasand Otmar Venjakob for their efforts to make my stay in Heidelberg a very pleasant time and for their help when writing andimproving these notes. All computations were done in Sage [Sag12].

1 Iwasawa theory of the class group

Let F be a number field and let p be an odd prime. Suppose we are given a tower of Galois extensionsF = F(0) ⊂ F(1) ⊂ F(2) ⊂ ·· · such that the Galois group of F(n)/F is cyclic of order pn for all n> 1. WriteC(n) for the p-primary part of the class group of F(n) and write pen for its order.

Theorem 1 (Iwasawa 56 [Iwa59]). There exist integers µ , λ , ν , and n0 such that

en = µ pn +λ n+ν for all n> n0.

Christian WuthrichSchool of Mathematical Sciences, University of Nottingham, UK, e-mail: [email protected]

11

[email protected]

12 Christian Wuthrich

1.1 Zp-extensions

Let me first describe the tower of extensions that we are talking about. Set F(∞) =⋃

F(n). The extensionF(∞)/F is called a Zp-extension as its Galois group Γ is isomorphic to the additive group of p-adic integerssince it is the projective limit of cyclic groups of order pn. The most important example is the cyclotomic Zp-extension: If F =Q, then the Galois group of Q(µpn)/Q is

(Z/pnZ)×, which is cyclic of order (p−1)pn−1.

So there is an extension Q(n−1)/Q contained in Q(µpn) such that Q(∞) is a Zp-extension of Q. For a generalF , the cyclotomic Zp-extension is F(∞) =Q(∞) ·F .

It follows from the Kronecker-Weber theorem that Q(∞) is the unique Zp-extension of Q. It would bea consequence of Leopoldt’s conjecture that the cyclotomic Zp-extension is the only one for any totallyreal number field, see [NSW00, Theorem 11.1.2]. For a general number field F , the compositum of allZp-extension contains at least Zr2+1

p in its Galois group where r2 denotes the number of complex placesin F . For an imaginary quadratic number field F , for instance, the theory of elliptic curves with complexmultiplication provides us with another interesting Zp-extension, the anti-cyclotomic Zp-extension. It canbe characterised as the only Zp-extension F(∞)/F such that F(∞)/Q is a non-abelian Galois extension.

Lemma 1 (Proposition 11.1.1 in [NSW00]). The only places that can ramify in F(∞)/F divide p and atleast one of them must ramify.

In the cyclotomic Zp-extension of F , all places above p are ramified and there are only finitely many placesabove all other places.

1.2 The Iwasawa algebra and its modules

Let O be a “coefficient ring”, for us this will always be the ring of integers in a p-adic field; so O = Zp istypical. There is a natural morphism between the group rings O

[Gal(F(n)/F)

], which allows us to form the

limitΛ = lim←−

nO[Gal(F(n)/F)

].

This completed topological group ring, called the Iwasawa algebra and also denoted by O[[Γ ]], is far betterto work with than the huge group ring O[Γ ].

Proposition 1. To a choice of a topological generator γ of Γ , there is an isomorphism from Λ to the ringof formal power series O[[T ]] sending γ to T +1.

The proof is given in Theorem 5.3.5 of [NSW00]. By the Weierstrass preparation theorem, an elementf (T ) ∈O[[T ]] can be written as a product of a power of the uniformiser of O times a unit of O[[T ]] and timesa distinguished polynomial, which, by definition, is a monic polynomial whose non-leading coefficientsbelong to the maximal ideal.

Let X (n) be a system of abelian groups with an action by Gal(F(n)/F). If there is a naturally defined normmap X (n+1)→ X (n), then we can form X = lim←−X (n) and consider it as a compact Λ -module. For instance theclass groups C(n) above have a natural norm map between them. Also lots of naturally defined cohomologygroups will have such a map, too. Suppose now O/Zp is unramified, otherwise the power of p below mustbe replaced by a power of the uniformiser of O.

Proposition 2. Let X be a finitely generated Λ -module. Then there exist integers r, s, t, m1, m2, . . . , ms, n1,n2, . . . nt , irreducible distinguished polynomials f1, f2, . . . ft , and a morphism of Λ -modules

X −→Λr⊕

s⊕i=1

Λ/

pmiΛ⊕

t⊕j=1

Λ/

fn jj Λ

whose kernel and cokernel are finite.

Proofs can be found in [Ser95], [NSW00, Theorem 5.3.8] or quite different in [Was97, Theorem 13.12]and [Lan90, Theorem 5.3.1]. The main reason is that Λ is a 2-dimensional local, unique factorisation do-main.

Overview of some Iwasawa theory 13

As the ideals f jΛ and the integers r,. . . , nt are uniquely determined by X , we can define the followinginvariants attached to X . The rank of X is rankΛ (X) = r. The µ-invariant is µ(X) = ∑

si=1 mi and the λ -

invariant is λ (X) = ∑tj=1 n j ·deg( f j). Finally, if r = 0,

char(X) = pµ(X) ·t

∏j=1

fn jj Λ

is called the characteristic ideal of X . If r = 0, s6 1 and all f j are pairwise coprime, then X →Λ/char(X)has finite kernel and cokernel.

Let us summarise the useful properties of Λ -modules in a lemma. Write XΓ (n) for the largest quotient of

X on which Γ (n) = Gal(F(∞)/F(n)) acts trivially.

Lemma 2. Let X be a Λ -module.

1. X is finitely generated if and only if X is compact and XΓ is a finitely generated Zp-module.2. Suppose X is a finitely generated Λ -module. Then X is Λ -torsion, i.e., the Λ -rank of X is 0, if and only

if XΓ (n) has bounded Zp-rank.

3. If XΓ (n) is finite for all n, then there are constants ν and n0 such that |X

Γ (n) |= pen with en = µ(X) · pn +λ (X) ·n+ν for all n> n0.

Proofs can be found in §5.3. of [NSW00]. Note that if X = Λ/ f for an irreducible f , then XΓ (n) is finite,

unless f is a factor of the distinguished polynomial ω(n) = (1+ T )pn − 1 corresponding to a topologicalgenerator of Γ (n).

1.3 Proof of Iwasawa’s theorem

I will sketch the proof of theorem 1 only in the simplified case when F has a single prime p above p andthat this prime is totally ramified in F(∞)/F . Let L(n) be the p-Hilbert class field of F(n), i.e., the largestunramified extension of F(n) whose Galois group is abelian and a p-group. By class field theory the Galoisgroup of L(n)/F(n) is isomorphic to C(n).

L(∞)

K

F(∞)

X

XΓ

L(0)

F

Γ

C

Set L(∞) =⋃

L(n), which is a Galois extension of F(∞) with Galois groupX = lim←−C(n). The action of Γ (n) on C(n) translates to an action of Γ on Xgiven by the following. Let γ ∈ Γ and x ∈ X . Choose a lift g of γ to theGalois group of L(∞)/F and set xγ = gxg−1. So X is a compact Λ -module.

Define K to be the largest abelian extension of F inside L(∞). Then L(0)

and F(∞) are contained in K. The maximality of K shows that the Galoisgroup of K/F(∞) must be equal to XΓ .

Since K/F(∞) is unramified and F(∞)/F is totally ramified at p, the inertiagroup I at a prime above p in K gives a section of the map from Gal(K/F)→Γ . Since KI = L(0), we have Gal(K/F(∞)) = C(0) =: C and it has a trivialaction of Γ on it. Hence C is isomorphic to XΓ . Replacing in this argumentF by F(n), we can also conclude that X

Γ (n)∼=C(n).

In particular, it is always finite. Hence X is a finitely generated torsionΛ -module and lemma 2 c) implies the theorem. ut

Iwasawa has given an example in [Iwa73] of a Zp-extension with µ(X)>0, however he conjectured that µ(X) = 0 whenever the tower is the cyclotomic Zp-extension. This wasshown to be true by Ferrero–Washington [FW79] when F/Q is abelian.

The above proof can also be used to show that if F = Q then the class group of Q(n) has no p-torsion.Conjecturally this may even be true for F =Q(µp)

+, see §1.8.


1.4 Stickelberger elements

Let K be an abelian extension of Q. By the Kronecker-Weber theorem, there is a smallest integer m suchthat K ⊂Q(µm) called the conductor of K. For each a ∈

(Z/mZ)× write σa for the image of a under the map(Z/mZ

)× ∼= Gal(Q(µm)/Q

)→ Gal(K/Q) = G. The Stickelberger element for K is defined to be

θK =− 1m ∑

16a<m(a,m)=1

a ·σ−1a ∈Q[G]

and the Stickelberger ideal is I = Z[G]∩θK Z[G]. It is not difficult to show that I = I′θK with I′ being theideal in Z[G] generated by all c−σc with (c,m) = 1, see [Was97, Lemma 6.9].

Theorem 2 (Stickelberger). The Stickelberger ideal I annihilates the class group of K.

This means that for any fractional ideal a and any integer c coprime to m, the ideal becomes principal afterapplying (c−σc)θK to it. It is important to note that this theorem does not say anything interesting whenK is totally real as then θK is a multiple of the norm NK/Q. Hence it will not give us information about theclass number of Q(µp)

+. For a quadratic imaginary field K, this is an algebraic version of the analytic classnumber formula for K, see the remark (b) after theorem 6.10 in [Was97].

Here is the idea of the proof, for details see [Was97, Theorem 6.10] or [Lan90, Theorem 2.4].

Proof. We consider only the case K = Q(µp) for some odd prime p. In each ideal class there is a primeideal q of degree 1, i.e., it is split above some prime `≡ 1 (mod p). Take the Dirichlet character χ modulo` of order p such that χ(a)≡ a(`−1)/p (mod q) for all a. Fix a primitive `-th root of unity ξ . The Gauss sumof χ is defined to be

Ga(χ) =− ∑u mod `

χ(u)ξu ∈Q(µp,µ`).

One can show that Ga(χ) ·Ga(χ) = ` and that we have Ga(χ)(c−σc) ∈Q(µp) for all c coprime to p. Finallya detailed analysis of the valuation of this Gauss sum at all primes above ` reveals that for any β ∈ Z[G]

such that βθK ∈ Z[G], we have that qβθK =(Ga(χ)

)βOK is a principal ideal in the ring of integers OK of

K.


1.5 p-adic L-functions

F(∞)

F(n)

Γ (n)

F

Γ

Q∆

G(n)

G

Consider the cyclotomic Zp-extension F(∞) of F = Q(µp) for some odd prime p.Write G(n) for the Galois group of F(n) = Q(µpn+1) over Q and G = lim←−G(n) =

Gal(F(∞)/Q). Then G ∼= ∆ ×Γ with ∆ = G(0) and Γ = Gal(F(∞)/F). We write γafor the image of σa in Γ . The cyclotomic character χ : G→ Z×p splits accordinglyinto the Teichmüller character ω : ∆ → Z×p and κ : Γ → 1+ pZp. So for any a ∈ Z×p ,the character ω sends σa to a (p− 1)-st root of unity with ω(a)− a ∈ pZp andκ(γa) = 〈a〉= a/ω(a).

For i ∈ Z/(p−1)Z, consider the projector

εi =1

p−1

p−1

∑a=1

ωi(a)σ

−1a ∈ Zp[∆ ]

to the ω i-eigenspaces. We split up the Stickelberger element θ (n) = θF(n) ∈Q[G(n)] for

the field F(n) into p−1 elements θ(n)i ∈Qp[Gal(F(n)/F)] defined by εi ·θ (n) = θ

(n)i ·εi;

explicitly

θ(n)i =− 1

pn+1 ∑16a<pn+1

p-a

a ·ω−i(a) · γ−1a ∈Qp[Gal(F(n)/F)] .

Lemma 3. If i 6= 1 then θ(n)i ∈Zp[Gal(F(n)/F)] and if i 6= 0,1, then θi =

(θ(n)i

)n>1 belongs to lim←−Zp[Gal(F(n)/F)]=

Λ . If i 6= 0 is even then θi = 0.

Recall that the generalised Bernoulli numbers for a Dirichlet character χ of conductor m are defined by

m

∑a=1

χ(a)t eat

emt −1=

∞

∑r=0

Br,χtr

r!.

An explicit computation [Was97, Theorem 7.10] links the elements θi to these Bernoulli numbers and thetraditional Bernoulli numbers Br. Recall that the Br,χ and Br also turn up as values of the complex L-functionL(s,χ) and the Riemann zeta-function [Was97, Theorem 4.2]. Hence we find the interpolation property.

Theorem 3. For any even integer r > 1, we have

κ1−r(θ1−r) =−(1− pr−1)

Br

r= (1− pr−1)ζ (1− r).

Furthermore, for any r > 1 and any even j 6≡ r (mod p−1), we have

κ1−r(θ1− j) =−

Br,ω j−r

r= L(1− r,ω j).

For any s∈Zp we can extend κs : Γ → 1+ pZp linearly to κs : Λ →Zp. The p-adic L-functions are definedto be Lp(s,ω j) = κs(θ1− j). To represent the p-adic L-function as a map χ 7→ χ(θ1− j) is analogue to Tate’sdescription of complex L-functions in his thesis; often these maps are written as measures on the Galoisgroup Γ .

Now Lp(s,ω j) is an analytic function in s and the existence of such a function satisfying the abovetheorem is equivalent to strong congruences between the values of L(s,ω j) for negative integers s. Forinstance, one can deduce the Kummer congruences [Was97, Corollary 5.14] from the theorem.

Leopoldt showed that Lp(1,ω j) satisfies a p-adic analytic class number formula involving the p-adicregulator, see [Was97, Theorems 5.18 and 5.24]. The p-adic L-function for j = 0 corresponding to θ1 is notin Λ , instead it has a simple pole at s = 1.

The above L-functions are in fact the branches of the p-adic zeta-function discovered by Kubota andLeopoldt. There are generalisations to a much larger class of L-functions: Suppose K is a totally real numberfield and F/K an abelian extension of degree prime to p. Let χ be a character of the Galois group of F/Kinto the algebraic closure of Qp and suppose that F is still totally real. Take O to be the ring Zp[χ] generated


by the values of χ . Then there is a p-adic L-function Lχ ∈ O[[Γ ]] such that κs(Lχ) = Lp(s,χ) satisfiesLp(1− r,χ) = L(1− r,χω−r) ·∏p|p(1−χω−r(p)N(p)r−1) for all r > 1. See for instance [Wil90].

1.6 The main conjecture

Let 3 6 i 6 p− 2 be an odd integer. Consider the projective limit X of the p-primary parts of the classgroups of F(n) =Q(µpn+1). Since ∆ acts on this Zp-module, we can decompose it into eigenspaces for thisaction. Let Xi = εi X , which is now a finitely generated torsion Λ = Zp[[Γ ]]-module. Hence it makes senseto talk about its characteristic ideal.

Theorem 4 (Main conjecture). The ideal char(Xi) is generated by θi for all odd 36 i6 p−2.

This was first proven by Mazur-Wiles in [MW84], then generalised to totally real fields by Wiles in [Wil90].These proofs use crucially the arithmetic of modular forms. Later a proof was found using the Euler systemof cyclotomic units, see [CS06] and the appendix in [Lan90].

This theorem has many implications (some of which were known before the conjecture was proved). Wecan split up the p-primary part C of the class group of Q(µp) into eigenspaces Ci = εiC

Theorem 5. For every odd 36 i6 p−2, the order of Ci is equal to the order of Zp/B1,ω−i .

Theorem 6 (Herbrand-Ribet [Ri]). For any odd 3 6 i 6 p−2, the character ω i appears in C/Cp if andonly if p divides the numerator of Bp−i.

1.7 Cyclotomic units

The p-adic L-function can also be constructed out of the following units. For each c coprime to m, theelement (ζ c

m− 1)/(ζm− 1) is a unit in Z[ζm] where ζm a primitive m-th root of unity, called a cyclotomicunit. On the one hand they are linked to the p-adic L-function as m varies in the powers of p; in fact thep-adic L-functions can be obtained as a logarithmic derivatives of the Coleman series associated to thecyclotomic unit. See Propositions 2.6.3 and 4.2.4 in [CS06]. On the other hand they are linked to the classgroup: When m is a power of p, the index of the group generated by the cyclotomic units and the roots ofunity in Q(µm) is equal to the class group order of Q(µm)

+ within the group of units in Z[ζm].The cyclotomic p-units ζ c

m−1 form an Euler system, see §3.2 in [Rub00], the appendix in [Lan90] and§5.2 in [CS06], due to the fact that they make the Euler factors of the L-function appear in their compatibilitywith respect to the norm map:

NQ(µm`)/Q(µm)(ζm`−1) = (1−σ−1` )(ζm−1)

for any prime ` - m. These special elements provides a powerful way of bounding the class group in termsof values of the p-adic L-function and yield a proof of the main conjecture.

1.8 Vandiver’s conjecture

The theory so far only covered the minus part of the class group, i.e., Ci for odd i. Note that ⊕i evenCi is thep-primary part of the class group of Q(µp)

+.

Conjecture 1 (Vandiver). The class number of Q(µp)+ is not divisible by p.

Although one may argue (see end of §5.4 in [Was97]) that it is not likely to hold for all p, it is known tohold for all primes p6 39 ·222 see [BH11]. Moreover for all these 9163831 primes, the components Ci arealways cyclic of order p and there are at most 7 non-trivial components. However, probably there are primeswith Ci of order larger than p and probably the λ -invariant can get arbitrarily large.

It is known since Kummer that if p divides the class number of Q(µp)+ then p divides |Ci| for some odd

i, see Corollary 8.17 in [Was97].


Proposition 3. If Vandiver’s conjecture holds for p, then Ci is isomorphic to Zp/B1,ω−i for all odd i. More-

over C(n)i is a cyclic Zp[Gal(Fn/F)]-module for all n.

This is shown in Corollary 10.15 in [Was97].

Conjecture 2 (Greenberg [Gre01]). If F is totally real, then X = lim←−C(n) is finite.

1.9 Examples

Let us first take p = 5 and so i = 3 is the only interesting value. We take γ1+p to be the generator of Γ

corresponding to T +1. Then

θ(4)3 =2+2 ·5+52 +3 ·53 +4 ·54 +O(55)+

(4+4 ·5+52 +4 ·53 +O(54)

)·T

+(1+5+4 ·52 +O(54)

)·T 2 +O(T 3)

which is congruent to θ3 modulo ω(4) = (1+ T )54 − 1; in particular the above expression is the correctapproximation for the 5-adic L-function θ3. It is a unit in Λ as the leading term −B1,ω2 = 2+2 ·5+ · · · is a5-adic unit. Of course this is not surprising as the class group of Q(µ5) is trivial. So here X = 0 and en = 0for all n.

Now to the first irregular prime p = 37. Here the Bernoulli number B32 is divisible by 37. Accordingly,we expect a non-trivial ω5 part in the class group of Q(µ37). Indeed the approximation to the 37-adicL-function is

θ(3)5 =14 ·37+33 ·372 +13 ·373 +O(374)+

(16+6 ·37+32 ·372 +O(373)

)·T

+(29+9 ·37+13 ·372 +O(373)

)·T 2 +O(T 3).

This is not a unit as −B1,ω−5 is divisible by 37. From the fact that the second coefficient is a unit, weconclude that θ5 is a unit times a linear factor. Hence X is a free Z37-module of rank 1 and en = n+ 1 forall n. The fact which underlies the proof of Ribet’s theorem is that the Eisenstein series

G =− B32

2 ·32+ ∑

n>1∑d|n

d31qn

= 770932104121732640 +q+2147483649q2+617673396283948q3+4611686020574871553q4+···

of weight 32 is congruent modulo one of the primes above 37 in Q(µ12) to the cuspform

f = q+ζ12 q2 +(−ζ

312 +ζ

212−ζ12

)q3−ζ

212 q4 +

(2ζ

312 +ζ

212−2ζ

212−2

)q5 + · · ·

of weight 2 for the group Γ1(37) and character ω30.

2 Iwasawa theory for elliptic curves

2.1 Examples

Let Q(∞)/Q be the cyclotomic Zp-extension of Q and let E/Q be an elliptic curve. The theorem of Mordell-Weil shows that the group E(Q(n)) is finitely generated for all n. Is this still true for E(Q(∞))? In particularis the rank of E(Q(n)) bounded as n grows? The analogy with the case of global function fields suggeststhat this should be the case.

There is a second interesting group attached to E. For any elliptic curve E over a number field F , theTate-Shafarevich group X(E/F(n)) is a certain torsion abelian group whose definition we give in §2.2. Wewrite X(n) for its p-primary part which is conjectured to be finite for all n. The first four examples werecomputed with the methods in [SW13].


2.1.1 Example 1

Let E be the elliptic curve given by

E : y2 + xy = x3 − 6511x − 203353

which has E(Q) =X(E/Q) = 0 and it is labelled 174b2 in Cremona’s table [Cr]. It has bad reduction at 2(additive), 3, and 29 (both split multiplicative).

If p = 5 then the rank of E(Q(n)) is zero for all n and the group X(n) is trivial, too. Since a p-torsiongroup can not act with a single fixed point on a p-primary group, we have that E(Q(n)) has no p-torsion forall n.

2.1.2 Example 2

Let us take the same curve but now with p = 7. Then the rank will still be zero for all n. However if|X(n)| = pen , then en = pn +2n−1 for all n > 0. So the Tate-Shafarevich group will explode in this case.Note that this curve has a 7-isogeny defined over Q and one Tamagawa number is 7 and the number ofpoints in the reduction over F7 has 7 points. So p = 7 appears in various places. In fact X(n) is formed ofpn−1 copies of Z/pZ and two copies of Z/pnZ.

2.1.3 Example 3

Again with the same curve, but this time for the prime p = 13. Once more the rank remains 0 in the tower,however the p-primary part of X(E/Q(n)) grows with

en =⌊ p

p2−1pn− n

2

⌋for all n. This formula is shown in [Ku]. For instance e0 = e1 = 0, e2 = 12, e3 = 168, e4 = 2208, . . . Visiblythe growth does not obey the same type of regularity as in the previous examples. The difference is that Ehas supersingular reduction at p = 13.

2.1.4 Example 4

Let us consider now the curve 5692a1

E : y2 = x3 + x2 − 18x + 25

which has E(Q) = Z(0,5)⊕Z(1,3). For p = 3, one can show that the rank is 6 over Q(1) and it is 12 for allQ(n) with n> 2. The 3-primary part of X(E/Q(n)) is trivial for all n. Note however that we do not know ifX(E/Q) is finite or not.

2.1.5 Example 5

Finally, consider the curve 11a3E : y2 + y = x3 − x2

and consider the anti-cyclotomic Z3-extension above F = Q(√−7). The construction of Heegner points

allows us to produce points of infinite order P(n) ∈ E(F(n)). The tower of points is compatible in the sensethat the trace of P(n) to the layer below is (−1) ·P(n−1). It can be shown that these points and their Galoisconjugates generate a group of rank pn in E(F(n)). Hence this is an example in which the rank is notbounded. See [Ber01].


2.2 Selmer groups

Let E/F be an elliptic curve over a number field F . Set Σ to be the finite set of places in F consisting of allplaces above p, all places of bad reduction for E and all infinite places.

For any field K, we write H i(K, ·) for the group cohomology of continuous cochains for the profiniteabsolute Galois group Gal(K/K). The notation H i

Σ(F, ·) will stand for the cohomology for the Galois group

GΣ (F) of the maximal extension of F that is unramified outside F , see [NSW00, §8.3]; it can also bedescribed as the étale cohomology group H i

ét(Spec(OF)\Σ , ·) for the corresponding étale group scheme. Ifthe Galois module M is finite p-primary, then H i

Σ(F,M) is finite, see [NSW00, Theorem 8.3.19]. If M is a

finitely generated Zp-module then so is H iΣ(F,M), see [Rub00, Proposition B.2.7]. For any abelian group

A, we will denote the Pontryagin dual HomZ(A,Q/Z) by A.For any finite extension K/F , we define the Tate-Shafarevich group X(E/K) to be the kernel of the

localisation mapH1(K,E)→∏

vH1(Kv,E)

where the product runs over all places v of K and Kv denotes the completion at v. The non-trivial elementsin X(E/K) have an interpretation as curves of genus 1 defined over K with Jacobian isomorphic to E andwhich are counter-examples to the Hasse principle, see [Mil06, §17]. It is known that X(E/K) is a torsionabelian group such that the Pontryagin dual of the p-primary part is a finitely generated Zp-module forevery prime p. It is conjectured that X(E/K) is finite.

Let m be a power of p. For any extension K of F , the long exact sequence of cohomology for E[m] givesthe Kummer exact sequence

0 //E(K)/mE(K)

κ //H1(K,E[m]

)//H1(K,E)[m] //0.

For any finite extension K of F , we define the m-Selmer group Selm(E/K) as the elements in H1(K,E[m])that restrict to elements in the image of the local Kummer map κv : E(Kv)/pkE(Kv)→H1(Kv,E[m]) for allplaces v of K. This contains naturally E(K)/mE(K) as a subgroup whose quotient is X(E/K)[m]. Since allcocycles in the Selmer group are unramified outside Σ , we get an exact sequence

0 // Selm(E/K) // H1Σ

(K,E[m]

)// ⊕v∈Σ

H1(Kv,E)[m].

In particular, this shows that Selm(E/K) is finite. We can now form the two limits, induced by the inclusionand the multiplication by p map between E[pk] and E[pk+1]. We set

S(E/K) = lim−→k

Selpk(E/K) ⊂ H1

Σ (K,W ) and

S(E/K) = lim←−k

Selpk(E/K) ⊂ H1

Σ (K,T )

where T = lim←−kE[pk] is the (compact) p-adic Tate module and W = lim−→k

E[pk] = E[p∞] is the (discrete)p-primary torsion of E. It is true that lim←−H1

Σ

(K,E[pk]

)= H1

Σ(K,T ) by an argument of Tate, see [NSW00,

Corollary 2.3.5].The corresponding limit version of the Mordell-Weil group are lim−→E(K)/pkE(K) and lim←−E(K)/pkE(K).

The first can be seen to be equal to E(K)⊗ZQp/Zp , which is isomorphic to a direct sum of rank(E(K)) copies

of Qp/Zp . The latter is equal to E(K)⊗ZZp, which is equal to the sum of rank(E(K)) copies of Zp plus thefinite group E(K)[p∞]. By passing to the limits, we find the exact sequences

0 // E(K)⊗Qp/Zp// S(E/K) //X(E/K)[p∞] // 0

0 // E(K)⊗Zp // S(E/K) // lim←−kX(E/K)[pk] // 0


where the lower sequence remains exact because we have taken projective limits of finite groups. The groupon the right hand side of the second line is a free Zp-module which is conjecturally trivial. The first linecombines nicely the rank information with the Tate-Shafarevich group. The Pontryagin duals of the first lineand all the groups in the second line are finitely generated Zp-modules.

Later in §4, we will give another description of S(E/K) which does not use the Kummer map, but usesthe modules W and T only.

2.3 Iwasawa theory for the Selmer group

Given a Zp-extension F(∞)/F , we consider the limit S(E/F(∞)) = lim−→nS(E/F(n)) and its dual

X = S(E/F(∞)) = lim←−n

S(E/F(n)) (1)

which is naturally a compact Λ -module. The maps are induced by the natural inclusion E(F(n)) →E(F(n+1)) and the restriction map on the Tate-Shafarevich groups. Hence, if the Mordell-Weil group sta-bilises after a few steps, as in all but the last example above, then X will contain a Zp-module of thisrank. The other natural limit lim←−n

S(E/F(n)) with respect to the corestriction map is less interesting: Ifthe Mordell-Weil group stabilises, meaning that E(F(∞)) = E(F(n)) for some n, and the Tate-Shafarevichgroups are finite, then this limit is trivial.

Lemma 4. The Selmer group X is a finitely generated Λ -module for any Zp-extension.

Proof. We should show by lemma 2 that XΓ is a finitely generated Zp-module; this is the dual of the Γ -fixed part of S(E/F(∞)). Later in theorem 11, we will show that this is not too far from the dual of S(E/F),which is a finitely generated Zp-module.

Conjecture 3 (Mazur [Maz72]). If E has good ordinary reduction at all places in F above p and F(∞)/F isthe cyclotomic Zp-extension, then the Selmer group X is a torsion Λ -module.

Note that this conjecture implies, by proposition 2, that the largest free Zp-module in X has finite rank λ (X),so by the above this will imply that the rank of E(F(n)) stabilises. Moreover we have:

Proposition 4. If the conjecture holds then E(F(∞)) is a finitely generated Z-module. Suppose that X(E/F(n))[p∞]is finite for all n, then there are constants µ , λ , ν , n0 such that if

∣∣X(E/F(n))[p∞]∣∣ = pen , then en =

µ pn +λ n+ν for all n> n0.

Proof. The first part is Theorem I.5 in [Gre]. For the second part, we will have to show that XΓ (n) is very

close to the dual of S(E/F(n)) for all n. This is done in the control theorem 12 below.

As shown by the examples 2.1.4 and 2.1.5, none of the two assumptions in Mazur’s conjecture can beremoved. Here are two important result in support of the conjecture.

Theorem 7 (Mazur [Maz72]). If E(F) and X(E/F)[p∞] are finite, then the conjecture holds.

The main result of Kato in [Ka] implies the following.

Theorem 8. Let E be an elliptic curve defined over Q and let F be an abelian extension of Q. Supposethat E has good ordinary reduction at p, then the Selmer group for the cyclotomic Zp-extension is a torsionΛ -module.

2.4 Mazur-Stickelberger elements

Let E/Q be an elliptic curve. We will suppose that E has good reduction at p. Let ωE be a Néron differentialon E; this is just dx

2y when E is given by a global minimal model. The canonical lattice ZE for E is the image


of∫

: H1(E(C),Z

)→C sending a closed path γ on E(C) to

∫γ

ωE . We define ΩE to be the smallest positiveelement of ZE .

The theorem of modularity [BCDT01] shows that there exists a morphism ϕE of curves X0(N)→ Edefined over Q. We take one of minimal degree. If f is the newform corresponding to the isogeny classof E, then there is a natural number cE , called the Manin constant, such that cE ·ϕ∗E (ωE) is equal to thedifferential 2πi f (z)dz on X0(N) corresponding to f , written here as a differential in the variable z on theupper half plane H. For the so-called optimal curve in the isogeny class one expects cE = 1.

For any rational number r = am , consider the ray from r to i∞ in the upper half plane. Its image in

X0(N)(C) is a (not necessarily closed) path r,∞ between two cusps.

Proposition 5 (Manin [Ma]). There is a natural number t > 1 such that, for all r ∈Q, the value of λ f (r) =2πi

∫ ri∞ f (z)dz belongs to 1

t ZE .

This is clear for the closed paths, i.e., when r is Γ0(N)-equivalent to i∞. The proof for general r uses theHecke operators T` on X0(N).

We define the (plus) modular symbol [r]+ by

[r]+ =1

ΩE·Re(

2πi∫ r

i∞f (z)dz

)∈Q,

see [Cr] for more details. For an abelian field K of conductor m, we define the Stickelberger element for Eto be

ΘE/K = ∑16a<m(a,m)=1

[ am

]+σa ∈Q

[Gal(K/Q)

].

Let ` be a prime of good reduction. The `-th coefficient a` of f satisfies `−a`+1 = #E(F`). If ` does notdivide m, then

NQ(µm`)/Q(µm)

(ΘE/Q(µm`)

)= (−σ`)

(1−a` σ

−1` +σ

−2`

)(ΘE/Q(µm)

),

which can be deduced from the action of the Hecke operator T` on X0(N).

2.5 The p-adic L-function

Let p be a prime of good reduction for E. Write now Θ(n)E for the Stickelberger element for the field Q(n)

and write G(n) = Gal(Q(n)/Q). We define the map j : Q[G(n)]→ Q[G(n+1)] to send an element of G(n) tothe sum over all its preimages in G(n+1). Then the norm N : Q[G(n+1)]→Q[G(n)] sends Θ

(n+1)E to

N(Θ

(n+1)E

)= ap ·Θ (n)

E − j(Θ

(n−1)E

).

This is shown using the Hecke operator Tp. Let α be a root of the polynomial X2−ap X + p. We set

L(n)E =

1αn+1 ·Θ

(n)E −

1αn+2 j

(Θ

(n−1)E

)(2)

for all n> 1. Then LE = (L(n)E )n>1 belongs to lim←−Q[Gal(Q(n)/Q)] and it is called the p-adic L-function of

E. Explicitly, we have

L(n)E = ∑

16a<pn+1

p-a

(1

αn+1

[ apn+1

]+− 1

αn+2

[ apn

]+)· γa.

Suppose now that E has good ordinary reduction at p. Then ap is a p-adic unit and hence we can find oneroot α which belongs to Z×p . Because the denominator of [ a

m ] is uniformly bounded, LE actually belongs toQp⊗Λ and in many cases it is known that LE ∈ Λ . For the supersingular case there is no unit root α andLE will never belong to Λ , see [Pol03].


Theorem 9. The p-adic L-function satisfies the interpolation properties

1(LE) =(

1− 1α

)2· L(E,1)

ΩE(3)

and

χ(LE) =Ga(χ)αn+1 ·

L(E, χ,1)ΩE

(4)

for all character χ of conductor pn+1 on Γ , i.e, that factor through Gal(Q(n)/Q) but not throughGal(Q(n−1)/Q).

The proof connecting the corresponding finite sums of modular symbols to the Mellin transform of themodular form can be found in formula (8.6) and §14 of [18].

Theorem 10 (Rohrlich [Roh84]). Only finitely many of the values χ(LE) in equation (4) are zero. Inparticular LE 6= 0.

Again, we can define the analytic function Lp(E,s) = κs−1(LE). If L(E,1) 6= 0, we know that E(Q)is finite. As a consequence of the above theorem, we see that Lp(E,1) 6= 0 as α 6= 1. Moreover the valueLp(E,1) is then predicted by the Birch and Swinnerton-Dyer conjecture.

Conjecture 4 (p-adic version of the Birch and Swinnerton-Dyer conjecture [18]). The order of vanishingof Lp(E,s) at s = 1 is equal to the rank of E(Q). The leading term of its series at s = 1 is equal to(

1− 1α

)2·

Regp(E/Q) ·#X(E/Q) ·∏v cv(#E(Q)tors

)2

where cv are the Tamagawa numbers and Regp(E/Q) is the p-adic regulator, see §3.2.

It would be very interesting to know that the order of vanishing of Lp(E,s) is equal to the order of vanishingof L(E,s). However this is only known when the p-adic L-function vanishes to order at most 1 by [PR87].

2.6 The main conjecture

Let E/Q and suppose E has good ordinary reduction at p. By theorem 8, we can associate to E the charac-teristic ideal char(X) of the dual of the Selmer group in (1). Under the same hypothesis, we have constructeda p-adic L-function (2).

Conjecture 5 (main conjecture). The p-adic L-function LE is a generator for the characteristic idealchar(X).

The other series of lectures will talk about the main result by Skinner and Urban on this conjecture.See [Wan] in these proceedings.

The generalisations to higher weight modular forms for Γ0(N) with p - N and p - ap is fairly straightforward, see [18]. For the extensions to p | N, but p2 - N, one has to be a bit careful as the case of splitmultiplicative reduction behaves differently due to the presence of exceptional zeroes because α = 1. Fi-nally the generalisation to the supersingular case is clearly much more complicated. To my knowledge thegeneralisation to additive reduction, i.e., when p2 | N, is not yet fully done.

3 The leading term formula

3.1 Control theorem

As before let E/F be an elliptic curve and let F(∞)/F be a Zp-extension. Recall that X is the dual of thelimit Selmer group S(E/F(∞)) as defined in (1) and we are interested in comparing XΓ with the dual of


S(E/F). For a place v ∈ Σ , write J(∞)v = ∏v|w H1(F(∞)

w ,E)[p∞]. We have the following diagram

0 // S(E/F(∞))Γ // H1Σ(F(∞),W )Γ // ⊕

v∈Σ

(J(∞)

v)Γ

0 // S(E/F)

α

OO

// H1Σ(F,W )

β

OO

// ⊕v∈Σ

H1(Fv,E)[p∞].

⊕vγv

OO(5)

We want to bound the kernel and cokernel of α . Even if it is clear that E(F(∞))Γ = E(F), it is not obviouswhat happens to the map E(F)⊗Qp/Zp →

(E(F(∞))⊗Qp/Zp

)Γ as a non-divisible point in E(F) can becomedivisible by p in E(F(n)).

Theorem 11. The kernel of α is finite and the dual of the cokernel is a finitely generated Zp-module.

Proof. The inflation-restriction sequence [NSW00, Proposition 1.6.6] for the H iΣ

-cohomology of W =

E[p∞] gives that the kernel of β is H1(Γ ,H0

Σ(F(∞),W )

)and the cokernel lies in H2

(Γ ,H0

Σ(F(∞),W )

). Now

the dual of D = H0(F(∞),W ) = E(F(∞))[p∞] has Zp-rank at most 2. Hence the dual of the exact sequence

0 //DΓ //Dγ−1 //D //DΓ

//0

shows that H1(Γ ,D) = DΓ has the same corank as DΓ = E(F)[p∞], which is finite. Hence the kernel ofβ and α are finite. In fact, if D is finite as in almost all cases, then the kernel of β has the same order asE(F)[p∞].

The cokernel of β is trivial, because H2(Γ ,D) = lim−→kH2(Γ ,E(F(∞))[pk]

)and the latter groups are trivial

because Γ has cohomological dimension 1, see [NSW00, Proposition 1.6.13]. Since β is surjective we seethat the duals of H1

Σ(F(∞),W )Γ and S(E/F(∞))Γ are finitely generated Zp-modules.

Note that this proves lemma 4 saying that X is a finitely generated Λ -module.

Theorem 12 (Control theorem). Suppose that E has good ordinary reduction at all primes that ramify inF(∞)/F. Then the map α has finite kernel and cokernel.

Proof. From the previous proof, we see that we are left to show that the cokernel of α is finite. By thesnake lemma applied to (5), it suffices to show that the kernel of ⊕γv is finite under our hypothesis.

Let v ∈ Σ . By local Tate duality [Tat95b], the group H1(Fv,E)[p∞] is the Pontryagin dual of the p-adic completion E(Fv)

? = lim←−E(Fv)/pkE(Fv) of the local points. The structure of elliptic curves over localfields, see chapter 7 in [Sil09], can be used to show that E(Fv)

? is finite if v - p and it is a finitely generatedZp-module of rank [Fv : Qp] if v | p. Choose a place w above v in F(∞). We wish to show that the kernel of

γv : H1(Fv,E)[p∞] //H1(F(∞)w ,E)[p∞]

is finite. Again by Tate duality this map is dual to the norm map

γv : E(F(∞))? //E(Fv)?.

The following lemmas will conclude this proof.

We will write x ×= y if there is a p-adic unit u such that x = u · y.

Lemma 5. If v splits completely in F(∞)/F, then kerγv = 0. Otherwise, if v is unramified, then #kerγv×= cv,

the Tamagawa number of E/Fv. In particular kerγv is non-zero for only finitely many v.

Proof. If v splits completely, then F(∞)w = Fv and the “norm” map is clearly surjective.

First assume v - p. The local extension F(∞)w /Fv is unramified and so the Néron model of E will not

change in this extension. Let Φ be its group of components and write E0v for the connected component of

the special fibre. Then we have that


0 //E0v (Fv)

? //E(Fv)? //Φ(Fv)

? //0

because the points in the formal group E are divisible by p when v - p. Now the norm map on the left handside is surjective because of Lang’s theorem [Lan56]. On the right hand side instead the norm map will bethe zero map for sufficiently large n. Hence kerγv is dual to Φ(Fv)

?.Now assume v | p, but the extension F(∞)/F is unramified at v. The argument is the same as above, except

that we now have to show that the norm is surjective on the formal groups. That is done in the part a) of thefollowing lemma.

Lemma 6. Let E be an elliptic curve over a p-adic field K and let L/K be a cyclic extension of degree apower of p. Let mL and mK be the maximal ideals of L and K respectively.

1. If L/K is unramified then the norm map on the formal group E(mL)→ E(mK) is surjective.2. If L/K is ramified and E has good ordinary reduction. Then the cokernel of the norm map on the formal

groups is finite.3. If v | p is totally ramified and E has good ordinary reduction, then #kerγv

×= N2

v where Nv is the numberof points in the reduction E(Fv).

Proof. For the proof of the first point uses the filtration by E(mk), the formal logarithm that gives anisomorphism E(mk

L)∼= 1+mk

L for large enough k, the fact that H1(FL/FK ,FL) = 0 for the residue fields,and the surjectivity of the norm map on units [Ser68, Proposition V.3].

The proof of the latter two can be found in [LR78]. It relies on the fact that the formal group of Ebecomes isomorphic to the multiplicative formal group over the ring of integers of the completion of themaximal unramified extension of Fv.

A different and more accessible proof of item 3 in the above lemma is explained in Lemma 4.6 in [Gre].It should be noted that both item 2 and item 3 are no longer valid when the reduction is supersingular. Thetheory in the case of good supersingular reduction at primes above p is quite different.

One can now add a proof for theorem 7. If E(F) and X(E/F)[p∞] are both finite, then so is S(E/F).By the control theorem 12, this implies that XΓ is finite. Since the Γ -coinvariants ΛΓ

∼= Zp of Λ are notfinite, we see that X is a torsion Λ -module. In fact, we see that this holds for all Zp-extensions, not just thecyclotomic. In example 2.1.5, the rank of E(F) is 1.

3.2 p-adic heights

We will now construct an analogue to the real-valued Néron-Tate height. We present a version in-spired by [Blo80]. Let E/F be an elliptic curve and we suppose that E has good ordinary reductionat all places above p that are ramified. To each cohomology class in H1(F,T ) represented by a cocycleξ : GF = Gal(F/F)→ T = Tp(E), we associate an extension

0 //Tpµ //Tξ//T //0

where Tpµ = lim←−µ[pk], also denoted by Zp(1), is a free Zp-module of rank 1 on which GF acts via thecyclotomic character. As a Zp-module Tξ = Tpµ⊕T , but the GF -action is given by

σ(ζ ,P) =(

σ(ζ ) · e(ξσ , σ(P)

), σ(P)

)for ζ ∈ Tpµ and P ∈ T ,

with e : T ×T → Tpµ denoting the Weil-pairing [Sil09, §3.8]. It is not hard to show that the class of the ex-tension Tξ does not depend on the chosen cocycle and that the boundary maps ∂ : H i(F,T )→H i+1(F,Tpµ)are given by applying the Weil-pairing to the cup-product with ξ , at least up to sign.

Consider now the commutative diagram with exact rows


0 // H1(F,Tpµ) //

H1(F,Tξ ) //

H1(F,T ) //

H2(F,Tpµ)

0 // ∏v H1(Fv,Tpµ) // ∏v H1(Fv,Tξ ) // ∏v H1(Fv,T )

∏∂v// ∏v H2(Fv,Tpµ)

(6)

with the product running over all places in F . It follows from global class field theory that the downwardsarrow on the right is injective [NSW00, Corollary 9.1.8.ii] if T is replaced by µpk ; that the limit is stillsurjective needs an additional argument [Tat76, Corollary to Proposition 2.2]. On the left we have the mapfrom the p-adic completion (F×)? of F× to ∏v(F×v )?.

Choose an topological generator γ . Let l : Γ → Zp be the map that send γ to 1. For each place v considerthe composition

λv : F×v //A×F //Γl //Zp

where A×F is the idèle group of F and the map that follows it is the reciprocity map. This map extends tothe completion λv : (F×v )? → Zp. In case F(∞)/F is the cyclotomic Zp-extension, then l is a multiple oflogp κ . For finite places v away from p, the map is simply λv(x) = (logp(κ(γ))

−1 · log(#Fv) · v(x) whereFv is the residue field. For places above p, we get λv(x) =−(logp(κ(γ))

−1 · logp(NKv/Qp(x)).Suppose now ξ belongs to S(E/F). Let S(E/F)0 be the subgroup of S(E/F) of all elements η such

that resv(η) ∈ E(Fv)? lies in the image of the norm from E(F(∞)

w )? for all places v. By lemma 5 and 6,this subgroup has finite index in S(E/F). Let η ∈ S(E/F)0. Since both resv(η) and resv(ξ ) belong tothe image of E(Fv)

? inside H1(Fv,T ), their cup-pairing is trivial. This is again a consequence of local Tateduality [Tat95b]. Hence resv(η) is sent to 0 by ∂v. By the injectivity of the right arrow in (6), we concludethat there is an element ζ in H1(F,Tξ ) that maps to η in H1(F,T ).

For each place v, we will define a local lift ζv ∈ H1(Fv,Tξ ). Since η ∈ S(E/F)0, there is an element

ηv ∈ H1(F(∞)w ,T ) whose norm is resv(η). Pick any lift of ηv to H1(F(∞)

w ,Tξ ) and define ζv to be its norm inH1(Fv,Tξ ).

By construction resv(ζ )− ζv ∈ H1(Fv,Tξ ) maps to 0 in H1(Fv,T ) and hence we can viewed it as anelement in H1(Fv,Tpµ) = (F×v )?. We set

〈ξ ,η〉= ∑v

λv(resv(ζ )−ζv

)∈ Zp.

It is not hard to check that this element is independent of the choices made, because both (F×)? and thenorms from H1(F(∞)

w ,Tpµ) lie in the kernel of ∑v λv.Since S(E/F)0 has finite index, we can linearly extend this to a pairing on S(E/F) with values in

Qp. This is called the canonical p-adic height pairing corresponding to the Zp-extension and the choice ofγ . Note that this construction only works under the assumption that E has good ordinary reduction at theramified places. For the generalisation to any Galois representation, which is de Rham at places above p,see [PR95, §3.1.2].

There is a variant of this construction: Let ξ ∈ S(E/F)0 and η = (η(n))n ∈ S(E/F(∞)), then one canconstruct in a similar way an element of Qp/Zp . This time one lifts resw(η

(n))∈H1(F(n)w ,W ) to H1(F(n)

w ,Tξ ⊗Qp/Zp) etc. It turns out that the map δ : S(E/F)0→Hom

(S(E/F(∞)),Qp/Zp

)= X has its image in XΓ . Now

the p-adic height pairing can be described involving the map π : XΓ → X → XΓ .

Proposition 6 (Proposition 3.4.5. in [PR92]). There is a commutative diagram

XΓ

α

XΓπoo

S(E/F)ι// // HomZp

(S(E/F),Zp

)S(E/F)0

hp

oo

δ

OO (7)

where hp is the p-adic height pairing and ι is a naturally defined surjective Zp-morphism with finite kernel.


Finally one should mention that the p-adic height pairing has also an analytic description using canonicalp-adic sigma-functions σv for all ramified places. These are well-explained in [MT91] and a fast algorithmfor computing them was found in [MST06] using Kedlaya’s algorithm. For instance, if E/Q and P= (x,y)∈E(Q) is a point that has good reduction everywhere and reduces to 0 at p, then hp(P) = logp(σp(P))−logp(e) where e is the square root of the denominator of x. In general the formula allows one to computethe p-adic height with only the information of E over F together with the explicit maps λv.

Conjecture 6 (Schneider [Sch82]). The canonical p-adic height pairing for the cyclotomic Zp-extension onan elliptic curve with good ordinary reduction at all places above p is non-degenerate.

Suppose X(E/F)[p∞] is finite. Choose a basis of E(F) modulo torsion and let Regγ(E/F) ∈ Qp bethe determinant of the p-adic height pairing on this basis. The number Regp(E/F) = Regγ(E/F) ·logp(κ(γ))

rankE(F) is independent of the choice of γ . The above conjecture then says that Regγ(E/F) 6= 0 inthe cyclotomic case. For the anti-cyclotomic Zp-extension it can well be that the p-adic height is degenerate.

3.3 Leading term

The following theorem was proved by Perrin-Riou for curves with complex multiplication then in generalby Schneider [Sch85]. See [PR93] for the details to complete our sketch of proof.

Let F(∞)/F be a Zp-extension such that all ramified places are totally ramified. Write Σ(ram) for the setof all the ramified places in F and denote by S the set of all places that split completely in F(∞).

As before, identify Λ with Zp[[T ]] via the choice of a topological generator γ . Let FX (T ) ∈ Zp[[T ]] be agenerator of the characteristic ideal for X .

Theorem 13. Suppose E has good ordinary reduction at all ramified places above p and suppose that thecanonical p-adic height for F(∞)/F is non-degenerate. Then

1. X is Λ -torsion;2. the characteristic series FX (T ) has a zero of order rankZp

(S(E/F)

)at T = 0;

3. if X(E/F)[p∞] is finite then the leading term F∗X (0) of FX (T ) at T = 0 satisfies

F∗X (0)×= ∏

v∈Σ(ram)

N2v ·

Regp(E/F) ·#X(E/F)[p∞] ·∏v6∈S cv·(#E(F)tors

)2 .

If the main conjecture holds for a curve E/Q, then the finiteness of X(E/Q)[p∞] and the non-degeneracyof the p-adic height pairing imply the p-adic BSD conjecture, up to a p-adic unit in the leading term. Thisis because 1−1/α

×= Nv. Theorem 13 together with Kato’s theorem 16 can be used to give a new efficient

algorithm [SW13] for the determination of the rank of E(Q) and upper bounds of X(E/Q)[p∞] for almostall p.

The proof of this theorem follows surprisingly closely what Tate [Tat95a] did in the function field case toreduce BSD to the finiteness of the p-primary part of the Tate-Shafarevich group. The algebraic part relieson the following lemma which can be deduced from proposition 2.

Lemma 7. Let X be a finitely generated Λ -module and suppose the cokernel of π : XΓ → XΓ is finite. Then

1. X is Λ -torsion;2. π has finite kernel;3. the leading term of its characteristic series satisfies F∗X (0)

×= #coker(π)

#ker(π) =: q(π).

If XΓ is finite, then q(π) is the Euler-characteristic ∏1i=0 #H i

(Γ ,S(E/F(∞))

)(−1)i. A proof in this case can

be found in §3 of [CS00] or in §4 of the chapter of [CGRR99] contributed by Greenberg.

Proof (Proof of theorem 13). From diagram (7) and the assumption that hp has finite kernel and cokernel,we find that π must have finite cokernel. Hence the lemma applies and we are left to determine the value ofq(π). Note that δ has now also finite kernel and cokernel.

Global duality [NSW00, Theorem 8.6.13] gives us a long exact sequence.


0 // S(E/F) // H1Σ(F,W )

((⊕v∈Σ

H1(Fv,E)[p∞]

vv0 H2

Σ(F,W )oo S(E/F)oo

Because X is Λ -torsion we have that lim←−S(E/F(n)) = 0 as shown in [PR92, §3.1.7]. When taking the limitof these above sequence over n, the resulting exact sequence is

0 //S(E/F(∞)) //H1Σ(F(∞),W ) //⊕

v∈Σ

J(∞)v //0

where J(∞)v = ∏w|v H1(F(∞)

w ,E)[p∞].Now we can produce the following big diagram with exact rows. As it is too long we write it in two parts,

the top two arrows on the right continue as the lower two arrows on the left.

0 // S(E/F(∞))Γ // H1Σ(F(∞),W )Γ // ⊕

v∈Σ

(J(∞)v )Γ // . . .

0 // S(E/F)

α

OO

// H1Σ(F,W )

β

OO

// ⊕v∈Σ

H1(Fv,E)[p∞]

⊕γvOO

// . . .

. . . // S(E/F(∞))Γ//

δ

H1Σ(F(∞),W )Γ

//

ε

0

. . . // S(E/F) // H2Σ(F,W ) // 0

To show that the two right squares commute requires some work [PR92, §4.4 and 4.5]. By the way, I amcheating slightly as in fact the term S(E/F) should be replaced by S(E/F)0 and other terms should bemodified accordingly. Also one has to show that (J(∞)

v )Γ = 0 to get the exactness in the top right corner; thisfollows from the triviality of H2(Fv,E)[p∞], again by local Tate duality.

Next, the transgression map ε is part of the short exact sequences from the degenerating Hochschild-Serrespectral sequence. It is injective and the cokernel is equal to H2

Σ(F(∞),W )Γ . However the group H2

Σ(F(∞),W )

is trivial as shown in §3.4.1 in [PR92]. This shows that ε is an isomorphism. So the big diagram gives theequality

q(α) ·q(β )−1 ·q(⊕

γv

)·q(δ ) = 1.

On the other hand the diagram (7) gives the equation

q(δ ) ·q(π) ·q(α) ·q(ι) = q(hp)

In the proof of theorem 11 we have seen that q(β ) ×=(#E(F)tors

)−1 if E(F(∞))[p∞] is finite, which followswithout too much difficulty from [LR78] under our assumptions. In lemma 5 and 6 we found that

q(⊕

γv

)×=

(∏v6∈S

cv · ∏v∈Σ(ram)

N2v

)−1

.

Finally, we know that E(F)⊗Zp has index #E(F)[p∞] in S(E/F) if the Tate-Shafarevich group is finite.Hence

q(hp)×=

Regp(E/F)

#E(F)tors.

It is not difficult to see that q(ι) = #X(E/F)[p∞] under our hypothesis. The neglected index of [S(E/F) :S(E/F)0] would have cancelled.


4 Selmer groups for general Galois representations

Let T be a free Zp-module of finite rank with an action of GF = Gal(F/F). We will suppose that onlyfinitely many places ramify in T ; so T has an action of GΣ (K) for a finite set Σ containing the places abovep and ∞. Let V = T ⊗Qp and W = V/T = T ⊗Qp/Zp . So far we were dealing with the example TE = TpEand WE = E[p∞] and VE is then a 2-dimensional Galois representation. But of course there are lots of otherexamples, such as more general subquotients of étale cohomology groups of varieties defined over F withQp-coefficients or Galois representation attached to modular forms.

We wish to define the Selmer group but we have no longer a Kummer map κ :?→ H1(Fv,W ) or ?→H1(Fv,T ). In order to understand how to define it in the general case, we look at how we could describe theimage of κ in the case of an elliptic curve.

4.1 Local conditions at places away from p

Let v∈ Σ be a prime not dividing p. Then the Kummer maps are E(Fv)⊗Qp/Zp→H1(Fv,WE) and E(Fv)?→

H1(Fv,TE). Recall that E(Fv) contains with finite index a group isomorphic to mv, the maximal ideal of Fv.Hence E(Fv)⊗Qp/Zp = 0 and E(Fv)

? = E(Fv)[p∞] =W GFvE is finite.

Here is the general definition for v - p. Define the subgroup H1f (Fv,V ) by the exact sequence

0 //H1f (Fv,V ) //H1(Fv,V ) //H1(I,V ) (8)

where I = Iv is the inertia subgroup in Gal(Fv/Fv). By the restriction-inflation sequence, H1f (Fv,V ) is isomor-

phic to H1(Furv /Fv,V I) where Fur

v is the maximal unramified extension of Fv. Then we define H1f (Fv,W ) as

the image of H1f (Fv,V ) under the map H1(Fv,V )→H1(Fv,W ). Also H1

f (Fv,T ) is the preimage of H1f (Fv,V )

from the map H1(Fv,T )→ H1(Fv,V ).For the case of the elliptic curve, we find H1

f (Fv,VE) = 0 for all v - p: In fact, we have in general thatH1

f (Fv,V ) = V I/(Frv−1)V I by Lemma 1.3.2 in [Rub00]. If the reduction is good then V IE = VE by the

Néron–Ogg–Shafarevich criterion [Sil09, Theorem 7.7.1] and Frv, the Frobenius of Gal(Furv /Fv), acts with

eigenvalues different from 1. If the reduction is multiplicative, then V IE∼=Qp(1) and the group H1

f (Fv,VE) isagain trivial. Finally if the reduction is additive, then V I

E = 0. Since we have the exact sequence

0 =V GFvE

//W GFvE

//H1(Fv,TE) //H1(Fv,VE) //H1(Fv,WE)

we obtain that H1f (Fv,WE) = 0 = E(Fv)⊗Qp/Zp and H1

f (Fv,TE) = E(Fv)[p∞] = E(Fv)?.

It would be tempting to define in general H1f (Fv,W ) without passing through V by replacing V with

W in (8). However the elliptic curve example explains that this does not work. In general, we have thatH1

f (Fv,W ) is the divisible part of the kernel of H1(Fv,W )→H1(Iv,W ) and H1f (Fv,T ) has finite index in the

corresponding kernel for T ; see [Rub00, Lemma 1.3.5].

4.2 Local conditions at places above p

Let now v be a place above p. The definition of the group H1f (Fv,V ) is given by Bloch-Kato [BK90] by

asking that0 //H1

f (Fv,V ) //H1(Fv,V ) //H1(Fv,V ⊗Bcris) (9)

is an exact sequence, where Bcris is a certain period ring of Fontaine.Now, if V is ordinary, one can give an easier definition. Here a general representation V is called ordinary

if there is a decreasing filtration Fili V of Gal(Fv/Fv)-stable subspaces of V such that the inertia group I actslike the i-th power of the cyclotomic character on the quotient Fili V/Fili+1 V . For an ordinary elliptic curve,


we consider the kernel of the reduction on E[pk] which is the pk-torsion E[pk] of the formal group. ThenFil1 VE = TpE⊗Qp sits in the middle between Fil2 VE = 0 and Fil0 VE =VE .

We set F+V = Fil1 V and then [Gre89] shows that

0 //H1f (Fv,V ) //H1(Fv,V ) //H1

(I,V /

F+V

)(10)

is exact. The subgroups H1f (Fv,W ) and H1

f (Fv,T ) are again defined as the image and preimage of H1f (Fv,V ),

respectively.

4.3 The Selmer groups

The Selmer groups are now defined as the following kernels. They are often denoted by H1f (F,W ) and

H1f (F,T ).

0 // S(W ) // H1Σ(F,W ) // ⊕

v∈Σ

H1(Fv,W )/H1

f (Fv,W ),

0 // S(T ) // H1Σ(F,T ) // ⊕

v∈Σ

H1(Fv,T )/H1

f (Fv,T )

.

They are now defined only in terms of T and equal to the previously defined Selmer group for elliptic curves.If X(E/F) is finite, then we have a way to determine the rank and the order of the Tate-Shafarevich groupfrom the Galois representation VE only. Note that the L-function L(E,s) is also constructed from VE only.

For instance, if T = Zp has a trivial GF -action on it, then S(Qp/Zp) is the dual of the p-primary part ofthe class group. If T = Zp(1) is of rank 1 with the action by GF given by the cyclotomic character, thenS(Qp/Zp(1)

)sits in a short exact sequence between O×F ⊗Qp/Zp and the p-primary part of the class group.

5 Kato’s Euler system

In this section we give a quick and very imprecise overview of the work of Kato in [Ka] where he proves onedivisibility in the main conjecture for elliptic curves (and modular forms of higher weight). See also [Sch98]and [8].

Let E/Q be an elliptic curve and p an odd prime at which E has good reduction. Let N be the conductorof E. We assume that E[p] is an irreducible GQ-module and hence all GQ-stable lattices in VpE are equalup to a scaling factor.

5.1 Construction of the Euler system

Let M be an integer. Pick an integer m> 5 coprime to 6M. For any elliptic curve A over a field k/Q, we canconstruct a division polynomial fm ∈ k(A)×, which is a function of divisor div( fm) =−m2(O)+∑P∈A[m](P)normalised such that [a]∗ fm = fm for all a coprime to m. Consider the maps g(m)

i for i = 1 or 2 that sendsa point in the modular curve Y (M) represented by (A,Q1,Q2) to fm(Qi). It is a rational function on Y (M)without zero, i.e., gi ∈ O(Y (M))×, called a Siegel unit. It can be shown that the function gi = g(m)

i ⊗1

m2−1 inO(Y (M))⊗Q is independent of the choice of m. They give rise to Beilinson element in K2

(O(Y (M))×

)⊗Q,

defined as the Steinberg symbol g1,g2.Such pairs of modular units can now be sent through a chain of maps. For a square-free r coprime to pN,

we take M = N pn+1 r and consider the map

X(M) //X(N)⊗Q(n)(µr)ϕE //E⊗Q(n)(µr).


We chase the pair of modular units through the maps (see §8.4 in [Ka])

O(Y (M)

)××O(Y (M)

)× ∪ // H2ét

(Y (N)/Q(n)(µr), Zp(2)

)twist à la Soulé

H2ét

(Y (N)/Q(n)(µr), Zp(1)

)

H1(Q(n)(µr),H1

ét(Y (N), Zp(1)

))

H1(Q(n)(µr), TE

)⊗Q

where we used that H1ét(E,Zp(1)

)= TE . Poincaré duality relates H1

ét(Y0(N),Qp

)to modular symbols as it is

equal to the homology H1(X0(N)(C),cusps,Z

)⊗Qp of paths between cusps. See §4.7 and §8.3 in [Ka].

The image of the Siegel units produce now elements c(n)r in H1Σ

(Q(n)(µr),T ) that form an Euler system

for a certain lattice T in TE ⊗Qp. See example 13.3 in [Ka] for details. In particular, (c(n)r )n belongs tolim←−n

H1(Q(n)(µr),T ). If ` is a prime not dividing rpN then they satisfy the norm relations

cor(c(n)r`

)=(

1− a``

σ−1` +

1`

σ−2`

)(c(n)r ).

See proposition 8.12 in [Ka] for the precise statement deduced from the Hecke operators on the modularcurves.

5.2 Relation to p-adic L-function

Suppose E has good ordinary reduction at p and let α ∈ Zp be the unit root of Frobenius. We continueto assume that E[p] is irreducible. The general “dual of exponential” à la Bloch-Kato has a very explicitdescription for elliptic curves. It is the map

exp∗ : E(Q(n)p )

?→(

E(Q(n)p )⊗Qp/Zp

)∧→Q(n)

p

which is a linear extension of the formal logarithm on the formal group with respect to the invariant differ-ential ωE . Based on the work of Coleman, Perrin-Riou has constructed an Iwasawa theoretic version whichinterpolates these maps:

Col : H1s :=

(E(Q(∞)

p )⊗Qp/Zp

)∧→Λ .

It is an injective Λ -morphism with finite cokernel such that for all character χ of Γ of conductor pn+1, wehave

χ(Col(z)

)=

Ga(χ)αn+1 ∑

σ∈Gal(Q(n)/Q)

χ(σ)exp∗(σ(z(n))

).

See the appendix of [Rub98] for details of the construction.One of the main theorems of Kato is

Theorem 14. There is an element c(∞) ∈ lim←−H1Σ(Q(n),T )⊗Q closely related to the ones constructed above

such that Col(c(∞)) = LE

References 31

This is theorem 16.6 in [Ka] with the “good choice” of γ+ as in 17.5. This is the technically most difficultpart of [Ka]. It implies that the Euler system is non-trivial by theorem 10. If the representation ρ : GQ →Gl(TE) is surjective, then c(∞) is integral by his theorem 12.5.4.

5.3 Euler system method

For each i, the limit Hi = lim←−nH i

Σ(Q(n),TE) is a finitely generated Λ -module, which does not depend on Σ

as long as it contains p. The existence of a non-trivial Euler system c(∞)r ∈ lim←−H1(Q(n)(µr),TE) proves the

following.

Theorem 15 (Theorem 12.4 in [Ka]).

1. H2 is Λ -torsion.2. H1 is a torsion-free Λ -module of rank 1.3. If E[p] is an irreducible GQ-module, then H1 is free of rank 1.

See also [Rub00]. The statement that H2 is Λ -torsion is called the weak Leopoldt conjecture and it isbelieved to hold for many Galois representations T . Global duality together with basic results deduced fromthe above theorem provides us with an exact sequence

0 //H1/c(∞)Λ

//H1s/

c(∞)Λ//X //H2 //0.

Here c(∞) ∈H1⊗Q is the part of the Euler system that is sent to the p-adic L-function LE by the Colemanmap; therefore Col sends the second term into Λ/LEΛ with finite cokernel. Hence the main conjecture isequivalent to

Conjecture 7. The characteristic ideal of H2 and of H1/c(∞)Λ are equal.

The advantage of this formulation is that it does not involve the p-adic L-function and makes sense in thesupersingular case as well.

Theorem 16 (Theorem 17.4 in [Ka]). Suppose E has good ordinary reduction at p. Then

1. X is a torsion Λ -module;2. there is an integer t > 0 such that the characteristic ideal char(X) divides pt LE Λ ;3. if the representation ρ : GQ→ Gl(TE) is surjective, then char(X) divides LE Λ .

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Tat76. John Tate, Relations between K2 and Galois cohomology, Invent. Math. 36 (1976), 257–274.Tat95a. John Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki, Vol. 9,

Soc. Math. France, 1995, pp. Exp. No. 306, 415–440.Tat95b. John Tate, WC-groups over p-adic fields, Séminaire Bourbaki, Vol. 4, Soc. Math. France, 1995, pp. Exp. No. 156,

265–277.Wan. Xin Wan, Skinner-urban lecutre series, To be updated with reference to itself.Was97. Lawrence C. Washington, Introduction to cyclotomic fields, second ed., Graduate Texts in Mathematics, vol. 83,

Springer-Verlag, New York, 1997.Wil90. Andrew Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), no. 3, 493–540.Wil95. Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551.

Introduction to Skinner-Urban’s Work on the Iwasawa MainConjecture for GL2

Xin Wan

1 Introduction

These notes are an introduction to the recent work of Christopher Skinner and Eric Urban [24] proving(one divisibility of) the Iwasawa main conjecture for GL2/Q (see Theorem 1). We give the necessary back-ground materials and explain the proofs. We focus on the main ideas instead of the details and therefore willsometimes be brief and even imprecise.

These notes are organized as follows. In Section 2 we formulate various Iwasawa main conjectures formodular forms. We also explain an old result of Ribet to illustrate the rough idea of the strategy behind thelater proofs. In Sections 3 and 4 we introduce the notions of automorphic forms and Eisenstein series onthe unitary group GU(2,2). Section 5 is devoted to explaining the Galois argument. Sections 6-9 give thetools used in the computation for the Fourier and Fourier-Jacobi coefficients of various Eisenstein series,which is a crucial ingredient in the argument. In Section 10 we give an example of a theorem of the authorgeneralizing the Skinner-Urban work.

2 Main conjectures

We introduce the objects required to state the Iwasawa main conjectures for GL2.

2.1 Families of Characters

Let p be an odd prime. Choose ι : C' Cp. Let GQ = Gal(Q/Q) and Q∞ ⊂Q(µp∞) be the cyclotomic Zp-extension of Q. Let ΓQ = Gal(Q∞/Q). Let ΛQ := Zp[[ΓQ]]. We also define ΛA = ΛQ,A = ΛQ⊗Zp A for A aZp-algebra. Let Ψ =ΨQ : GQ→Λ

×Q be the composition of GQ ΓQ with ΓQ →Λ

×Q . Let εQ be a character

of Q×\A×Q which is the composition of ΨQ with the reciprocity map of class field theory (normalized usinggeometric Frobenius elements). Take γ ∈ Γ to be the topological generator such that ε(γ) = 1+ p whereε is the cyclotomic character giving the canonical isomorphism Gal(Q(µp∞)/Q) ' Z×p . For each ζ ∈ µp∞

and integer k we let ψk,ζ be the finite order character of Q×\A×Q that is the composition of ΨQ with the mapΛ×Q → C×p that maps γ to ζ (1+ p)k. We also write ψζ for ψ0,ζ . We let ω be the Teichimuller character.

Institute for Advanced Study [email protected]

35

36 Xin Wan

2.2 Characteristic Ideals and Fitting ideals

Let A be a Noetherian normal demain and X a finite A-module. The characteristic ideal charA(X) ⊂ A isdefined to be zero if X is not torsion and

charA(X) = x ∈ A|ordPx≥ lengthAP(XP), for all height one primes P⊂ A.

Now take any presentationAr→ As→ X → 0

of X . The Fitting ideal is defined by the ideal of A generated by all the determinants of the s× s minors ofthe matrix representing the first arrow.

Remark 1. Fitting ideals respect any base change while characteristic ideals do not in general.

2.3 Selmer Groups for Modular Forms

Let f = ∑∞n=1 anqn ∈ Sk(N,ψ0), k ≥ 2, be a cuspidal eigenform with character ψ0 of (Z/NZ)× and let

L/Qp be a finite extension containing all Fourier coefficients an of f . Let OL be the ring of integers of L.Assume that f is ordinary, which means that ap is a unit in OL. Let ρ = ρ f : GQ→AutLVf be the usual twodimensional Galois representation associated to f . Then it is well known (by [14], for example) that thereis a GQp -stable L-line V+

f ⊂ Vf such that Vf /V+f is unramified. We fix a GQ-stable OL lattice Tf ⊂ Vf and

let T+f = Tf ∩V+

f .

Definition 1. (Selmer Groups) Let Σ be a finite set of primes.

SelΣL (Tf ) := kerH1(Q,Tf ⊗OL Λ∗OL

(Ψ−1)→ H1(Ip,(Tf /T+f )⊗OL Λ

∗OL

(Ψ−1))

× ∏6=p, 6∈Σ

H1(I`,Tf ⊗Λ∗OL

(Ψ−1))

where Λ ∗A =HomZp(ΛA,Qp/Zp) is the Pontryagin dual and Λ ∗OL(Ψ−1) means that the Galois action is given

by the character Ψ−1. LetXΣ

L (Tf ) := HomZp(SelΣL(Tf ),Qp/Zp)

andcharΣ

f ,Q( f ) = charΛQ,OL(XΣ

L (Tf )).


Let 0 < n < k−2 be an integer and ζ 6= 1 a pt−1th root of unity. Let Σ be a finite set of primes. We definethe algebraic part of a special L-value for f by:

LΣalg( f ,ψ−1

ζω

n,n+1) := ap( f )−tpt(n+1)n!LΣ ( f ,ψ−1

ζωn,n+1)

(−2πi)nτ(ψ−1ζ

ωn)Ωsgn((−1)m)f

where ap( f ) is the p-adic unit root of x2− apx+ pk−1ψ0 = 0, τ(ψ) is a Gauss sum for ψ and Ω±f are

Hida’s canonical periods of f . (There is also a formula for ζ = 1 which is more complicated which we omithere). The p-adic L-function is a certain element LΣ

f ,Q ∈ΛQ,OL characterized by the following interpolationproperty. Let φn,ζ : ΛOL → OL(ζ ) be the OL homomorphism sending γ to ζ (1+ p)n. Then:

φn,ζ (LΣf ,Q) = LΣ

alg( f ,ψ−1ζ

ωn,n+1),0≤ n≤ k−2.

This was constructed in [1], and also [18].

Introduction to Skinner-Urban’s Work on the Iwasawa Main Conjecture for GL2 37

2.5 The Main Conjecture

The Iwasawa main conjecture for f is the following

Conjecture 1. The module XΣL (Tf ) is a finite torsion ΛQ,OL -module and charΣ

f ,Q is generated by LΣf ,Q.

The main result that we are going to prove in this lecture series is:

Theorem 1. (Kato, Skinner-Urban) Suppose f has trivial character, weight 2 and good ordinary reductionat p. Suppose also that:

• The residual representation ρ f is irreducible.• For some p 6= `||N, ρ f is ramified at `.

Then the Iwasawa main conjecture is true in ΛQ,OL⊗Zp Qp. If, moreover, there exists an OL-basis of Tf withrespect to which the image of ρ f contains SL2(Zp), then the equality holds in ΛQ,OL .

The last condition is put by Kato who proved “⊇”. It is satisfied, for example, by the p-adic Tate modulesof semistable elliptic curves if p ≥ 11. We will focus on Skinner-Urban’s proof for “⊆”. The technicalcondition (ii) can be removed by working with forms over totally real fields and a base change trick .

2.6 Two and Three-Variable Main Conjectures

More p-adic characters: Let K/Q be an imaginary quadratic extension such that p splits as v0v0, where v0

is determined by our chosen isomorphism ι : C ' Cp. By class field theory there is a unique Z2p-extension

of K unramified outside p, which we denote by K∞. Let GK := Gal(K/K) and ΓK := Gal(K∞/K). Thereis an action of complex conjugation c on ΓK. We write Γ

±K for the subgroup on which c acts by ±1.

For any Zp-algebra A ⊂ Qp we define: ΛK,A = A[[ΓK]], Λ±K,A = A[[Γ±K ]] and generators γ± of Γ

±K by

requiring recKp((1+ p)12 ,(1+ p)±

12 ) = γ±. Here recKp is the reciprocity map of class field theory. (Note

that Kp ' Qp×Qp.) Let ΨK be the composition GK → ΓK → Λ×K,A. We define Ψ

±K similarly. We also

define characters εK,ε±K of K×\A×K by composing εK,ε±K with the reciprocity map.Now let f be a cuspidal eigenform of weight k ≥ 2. We define the set of arithmetic points by:

Xaf ,K := φ : OL homomorphism ΛK,OL → Qp : φ(γ+) = ζ

+(1+ p)k−2,φ(γ−) = ζ−,ζ± ∈ µp∞.

For φ ∈ Xaf ,K, let θφ := ω2−kχ

−1f ξφ with ξφ := (φ ΨK)(ε2−kεk−2.χ f ) and let fθφ

be the conductor of θφ .In particular we show that for any finite set Σ of primes containing all the bad primes (all the primes wheref or K is ramified), we have the two variable p-adic L-function LΣ

f ,K ∈ΛK,OL such that:

φ(LΣf ,K) = u f ap( f )

ordp(Nm(fθφ).((k−2)!)2g(θφ )Nm(fθφ

d)k−2LΣK( f ,θφ ,k−1)

(2πi)2k−2Ω+f Ω

−f

for any sufficiently ramified φ ∈ Xaf ,K, where d is the different of K and u f is a p-adic unit depending on

f . We remark that if ζ− = 1 then our special L-value is just the product of the special L-values for f andf ⊗χK twisted by some ψ such that ψ Nm = θφ . Here χK is the quadratic character for K/Q.We can also define Selmer groups SelΣ

f ,K and XΣf ,K, charΣ

f ,K in the exact same way as in the one-variablecase. We have the two-variable main conjecture:

Conjecture 2. XΣf ,K is a finite torsion ΛK,OL -module. Furthermore charΣ

K, f is principal and generated byLΣ

f ,K.

Additionally f can be embedded in a Hida family of ordinary cuspidal eigenforms f (we discuss these in thenext section in more detail). We can form a three-variable p-adic L-function LΣ

f,K and formulate a 3 variablemain conjecture.

38 Xin Wan

2.7 Comment on the Proof

The cyclotomic main conjecture for modular forms (conjecture 1) is deduced from a partial inclusion inthree-variable main conjecture. Roughly speaking the inclusion charf,K ⊆ (LΣ

f,K) in the three-variable mainconjecture, when specialized to the cyclotomic Zp-extension of K, implies that the inclusion char f ,Q.char f⊗χK,Q⊆(L f ,Q.L f⊗χK,Q) in the cyclotomic main conjecture for f and f ⊗ χK. By Kato’s work, this in turn impliesthat char f ,Q = (L f ,Q) and char f⊗χK,Q = (L f⊗χK,Q).We will focus on proving the inclusion charΣ

f,K ⊆ (LΣf,K) in the three variable main conjecture in the rest

of the paper. The general strategy for proving this inclusion is: the product of LΣK and the Σ -imprimitive

Kubota-Leopodlt p-adic L-function LΣ1,Q attached to the trivial character gives the congruences between

Eisenstein series and cusp forms on the unitary similitude group GU(2,2)⇒ the same congruences betweenreducible and irreducible Galois representations ⇒ required extension class in H1(K,−). The first arrowis by the Langlands correspondence and the second is a Galois theoretic argument, the so-called “latticeconstruction".

2.8 A Theorem of Ribet

In this section we review Ribet’s proof of the converse to Herbrand’s theorem [Ri]. This illustrates the mainideas in the strategy. In this section we set OL = Zp, λ the maximal ideal of OL, κ = OL/λ .

Theorem 2. Suppose j ∈ [2, p−3] is an even number. If p|ζ (1− j) then H1ur(GQ,F(ω1− j)) 6= 0 (the group

of everywhere unramified classes is non-zero).

Proof. For j 6= 2, we make use of the level 1 weight j Eisenstein series:

E j(q) =ζ (1− j)

2+ ∑

n≥1σ j−1(n)qn

where σ j−1(n) = ∑d|n d j−1. If p|ζ (1− j), then E j “looks” like a cusp form modulo p. We divide the proofinto three steps:

Step1: Construct a cusp form f ′ ∈ S j(SL2(Z),Zp) such that f ′ ≡ E j(mod p) (in terms of q-expansion). Thisis a case by case study using the fact that the ring of modular forms of level 1 is C[E4,E6].

Step 2: Prove that f ′ can be replaced by an eigenform f ∈ S j(SL2(Zp),OL) whose Hecke eigenvalues arethe same as those of E j modulo p. This can be proven by easy commutative algebra (essentially a lemma ofDeligne and Serre).

Step 3: The lattice construction: construct the class by comparing the Galois representations of E j andf . Note that the Galois representation for E j is ε j−1 ⊕ 1. It is easy to see that there is a σ0 ∈ Ip suchthat ε j−1(σ0) 6≡ 1(mod p). Since ap( f ) ≡ σ j−1(p) ≡ 1(mod p), f is ordinary. As we have noted before,

ρ f |GQp=

(αε j−1 ∗

α

)for some unramified character α . Take a basis v1,v2 such that

ρ f (σ0) =

(ε j−1(σ0)

1

).

Write ρ = ρ f and ρ(σ) =

(aσ bσ

cσ dσ

)for σ ∈ O[GQ].

Claim:

(a)aσ ,dσ ,bσ cτ ∈ O for σ ,τ ∈ OL[GQ] and aσ ≡ ω j−1(σ),dσ ≡ 1,bσ cτ ≡ 0(mod p);(b)C := cσ : σ ∈ OL[GQ] is a non-zero fractional ideal.(c)cσ = 0 if σ ∈ I` for all `.


Proof of the claim:

Let ε1 := 1ε j−1(σ0)−1 (σ0−1),ε2 := 1

1−ε j−1(σ0)(σ0− ε j−1(σ0)); one can check: ρ(ε1) =

(1 00 0

)and ρ(ε2) =(

0 00 1

). Thus aσ = traceρ(ε1σ) ∈ OL and traceρ(ε1σ) ≡ trace(ε j−1 + 1)(ε1σ) = ε j−1(σ). The claim for

dσ is proven similarly. Also since ρ(σ)ρ(τ) = ρ(στ), bσ cτ = aστ −aσ aτ ≡ 0(mod p).

(b) follows from the irreducibility of ρ and (c) can be seen from the description for ρ|GQpabove and the

triviality of ρ|I` for ` 6= p.

Let M1 = OLv1, M2 = Cv2, M = M1⊕M2 (which is easily seen to be the OL[GQ]-submodule generated byv1).

• M2 := M2/λM2 ' κ (note that C is non-zero by (b)) is a GQ stable submodule of M := M/λM. This isbecause for any m2 = cv2 ∈M2, ρ(σ)m2 = bσ cv1 +dσ cv2 ∈ λv1 +Cv2 by (a);

• by (a), GQ acts by ω j−1 and 1 on M1 = M/M2 and M2 respectively;• the extension: 0→ M2→ M→ M1→ 0 is non split since M is generated by v1 over OL[GQ].

Thus M gives a nontrivial extension class and it actually in H1ur(Q,κ(ω1− j)) by claim (c).

If j = 2 the Eisenstein series E2 is not holomorphic and we use Ep+1 in the place of E2.

3 Hermitian Modular Forms on GU(n,n)

3.1 Hermitian Half Space and Automorphic Forms

Let K/Q be an imaginary quadratic extension and let O be the ring of integers of K. Let GU(n,n) be the

unitary similitude group associated to the pairing(

1n−1n

)= ωn on K2n:

G := GU(n,n)(A) = g ∈ GL2n(O⊗A) : gωntg = λgωn,λg ∈ A×.

Here µ(g) := λg is the similitude character, and we write U :=U(n,n)⊂ G for the kernel of µ . We define

Q = Qn to be the Siegel parabolic subgroup of G consisting of block matrices of the form(

A BC D

)such that

C = 0. LetHn := Z ∈Mn(C) :−i(Z− tZ)> 0.

(Note that H1 is the usual upper half plane).

Let Z ∈Hn. For α =

(A BC D

)∈G(R) with A,B,C,D n×n block matrices. Let µα(Z) :=CZ+D,κα(Z) =

CtZ + D. We define the automorphy factor:

J(α,Z) := (µα(Z),κα(Z)).

Let G(R)+ = g ∈ G(R),µ(g)> 0 then G(R)+ acts on Hn by

g(Z) := (AgZ +Bg)(CgZ +Dg)−1, g =

(Ag BgCg Dg

).

Let K+∞ = g ∈U(R) : g(i) = i (we write i for the matrix i1n ∈ Hn) and Z∞ be the center of G(R). We

define C∞ := Z∞K+∞ . Then k∞ 7→ J(k∞, i) defines a homomorphism from C∞ to GLn(C)×GLn(C).

Definition 2. A weight k is a set of integers (kn+1, ...,k2n;kn, ...,k1) such that k1 ≥ k2 ≥ ... ≥ k2n and kn ≥kn+1 +2n.

40 Xin Wan

A weight k defines an algebraic representation of GLn(C)×GLn(C) by

ρk(g+,g−) := ρ(kn,...,k1)(g+)⊗ρ(−kn+1,...,−k2n)(g−)

where ρ(a1,...,an) is the dual of the usual irreducible algebraic representation of GLn with highest weight(a1, ...,an). Let Vk(C) be the representation of C∞ given by

k∞→ ρk J(k∞, i).

Fix K an open compact of G(A f ). We let

ShK(G) = G(Q)+\Hn×G(A f )/KC∞.

The automorphic sheaf ωk is the sheaf of holomorphic sections of

G(Q)+\Hn×G(A f )×Vk(C)/KC∞→ G(Q)+\H+n ×G(A f )/KC∞.

One can also define these Shimura varieties and automorphic sheaves in terms of moduli of abelian vari-eties. We omit these here.

The global sections of ωk is the space of modular forms consisting of holomorphic functions:

f : Hn×G(A f )→Vk(C)

which are invariant by some open compact K of the second variable, and satisfy:

µ(γ)k1+...+k2n

2 ρk(J(γ,Z))−1 f (γ(Z),g) = f (Z,g)

for all γ ∈ gKg−1∩G+(Q). Also, when n = 1 we require a moderate growth condition.

Remark 2. We will be mainly interested in the scalar-valued forms. In this case Vk(C) is 1-dimensional ofweight k = (0, ...,0;κ, ...,κ) for some integer κ ≥ 2.

3.2 Hida Theory

Hida Theory GL2/QWe choose a quick way to present Hida theory. Let M be prime to p and χ a character of (Z/pMZ)×.The weight space is SpecΛ for Λ := Zp[[T ]]. Let I be a domain finite over Λ . A point φ ∈ SpecI is calledarithmetic if the image of φ in SpecΛ is given by the Zp-homomorphism sending (1+T ) 7→ ζ (1+ p)κ−2

for some κ ≥ 2 and ζ a p-power root of unity. We usually write κφ for this κ , called the weight of φ . Wealso define χφ to be the character of Z×p ' (Z/pZ)××(1+ pZp) that is trivial on the first factor and is givenby (1+ p) 7→ ζ on the second factor.

Definition 3. An I-family of forms of tame level M and character χ is a formal q-expansion f=∑∞n=0 anqn,an ∈

I, such that for a Zariski dense set of arithmetic points φ the specialization fφ = ∑∞n=0 φ(an)qn of f at φ is

the q-expansion of a modular form of weight κφ , character χχφ ω2−κφ where ω is the Techimuller character,

and level M times some power of p.

Definition 4. The Up operator is defined on both the spaces of modular forms and families. It is given by:

Up(∞

∑n=0

anqn) =∞

∑n=0

apnqn.

Hida’s ordinary idempotent ep is defined by ep := limn→∞ Un!p . A form f or family f is called ordinary if

ep f = f or epf = f.

FACT The space of ordinary families is finite and free over the ring I.


Remark 3. For Hilbert modular forms the analogues space is still finite but not free in general. The subspaceof ordinary cuspidal families is both finite and free.

Hida Theory for GU(2,2)For simplicity let us restrict to the case when the prime to p part of the nebentypus is trivial. Fix some primeto p level group K of G(Z). Let T be the diagonal torus of U =U(2,2). Let χ be a character of T (Z/pZ).The weight space is SpecΛ2 where Λ2 is defined to be the completed group algebra of T (1+ pZ) = (1+pZ)4. Let A be any domain finite over Λ2.

Definition 5. A weight k = (k1,k2;k3,k4) is a set of integers ki such that k1 ≥ k2 +2≥ k3 +4≥ k4 +4.

Definition 6. A point φ ∈ SpecA is called arithmetic if its image in SpecΛ2 is given by the character [k]χφ .χwhere k is a weight and [k] is given by:

diag(t1, t2, t3, t4) 7→ tk31 tk4

2 tk23 tk1

4

(We identify U(Zp)'GL4(Zp) by the first projection of Kp 'Kv0×Kv0 ) and χφ is a finite order characterof T (1+ pZ).

We are going to define Hida families by a finite number of q-expansions: Let K ⊂ G(A f ) be a level groupX(K) be a finite set of representatives x of G(Q)\G(A f )/K with xp ∈ Q(Zp). For any g ∈ GU(2,2)(AQ)let S+

[g] comprise those positive semi-definite Hermitian matrices h in Mn(K) such that Trhh′ ∈ Z for all

Hermitian matrices h′ such that(

1 h′

1

)∈ NQ(Q)∩gKg−1.

Definition 7. For any ring A finite over Λ2 we define space of A-adic forms with tame level K ⊂G(A f ) andcoefficient ring A to be the elements:

F := Fxx∈X(K) ∈ ⊕x∈X(K)A[[qS+[x] ]]

such that for a Zariski dense set of arithmetic points φ ∈A the specialization Fφ of F at φ is the q-expansionof the matrix coefficient of the highest weight vector of holomorphic modular forms of weight kφ and

nebentypus χχφ ω(t−k31 t−k4

2 t−k23 t−k1

4 ).

(In the Skinner-Urban case the interpolated points φ are of scalar weights and thus do not need to take thehighest weight vector.)

Definition 8. Some Up operators: for t+ = diag(t1, t2, t3, t4) ∈ T (Qp) such that t2/t1, t3/t2, t3/t4 ∈ pZp. Wedefine an operator Ut+ on the space of Hermitian modular forms by: Ut+ . f = |[k∗](t)|−1

p f |kut+ where [k∗] =[k+(2,2;−2,−2)] and f |kut+ is the usual Hecke operator defined by double coset decomposition (with nonormalization factors).

Hida proved that this ut+ preserves integrality of modular forms and defined an idempotent:

eord := limn→∞

un!t+ .

A form or family F is called nearly ordinary if eordF = F . Again, we have that the space of nearly ordinaryHida families with coefficient ring A is finite and free over A. This is called the Hida’s control theorem forordinary forms.

Remark 4. In order to prove the finiteness and freeness (both in the GL2 and unitary group case) we needto go back to the notion of p-adic modular forms using the Igusa tower, which we omit here.

Another important input of Hida theory is the fundamental exact sequence proved [24, Chapter 6]. Welet C1(K) be the set of cusp labels of genus 2 and label K ([24, 5.4.3]). Write Λ1 for the weight ring ofU(1,1)/Q. Then Skinner-Urban proved the following

Theorem 3. For any Λ2-algebra A there is a short exact sequence

0→M0ord(K

p,A)→M1ord(K

p,A)→⊕[g]∈C1(K)M0ord(K

p1,g,Λ1)⊗Λ1 A→ 0.

42 Xin Wan

Here M0ord(K

p,A) is the space of A-valued families of ordinary cusp forms on GU(2,2), M1ord(K

p,A) isthe space of ordinary forms taking 0 at all genus 0 cusps ([24, 5.4]). The M0

ord(Kp1,g,Λ1) is the space of

ordinary cusp forms on U(1,1) with tame level group K p1,g for K p

1,g = GU(1,1)(A f )∩gKg−1 and GU(1,1)is embedded as the levi subgroup of the Klingen Parabolic subgroup of GU(2,2). The Φ is the “Siegeloperator” giving the restricting to boundary map. The Λ1-algebra structure for Λ2 is given by the embeddingT1 → T2 : (t1, t2)→ (t1,1, t2,1).

The proof is a careful study of the geometry of the boundary of the Igusa varieties ([24, 6.2,6.3]). Thistheorem is used to construct a cuspidal Hida family on GU(2,2) that is congruent to the Klingen Eisensteinseries modulo the p-adic L-function.

One more important property of ordinary families is that the specialization of a nearly ordinary family toa very regular weight is classical. This will be used to ensure that the ΛD-adic Hecke algebra of ordinaryΛD-adic form can not have CAP components.

4 Eisenstein Series on GU(2,2)

4.1 Klingen Eisenstein Series

Let P be the Klingen Parabolic subgroup of GU(2,2) consisting of matrices of the form× 0 × ×× × × ×× 0 × ×0 0 0 ×

.

Let MP be the levi subgroup of P defined by

MP ' GU(1,1)×ResOK/ZGm,(g,x) 7→

Ag Bg

µ(g)x−1

Cg Dgx

.

Let NP be the unipotent radical of P.

Observe that if π is an automorphic representation of GL2 and ψ is a Hecke character of A×K which restrictsto the central character χπ of π on A×Q, then these uniquely determine an automorphic representation πψ

of GU(1,1) with central character ψ . Now suppose we have a triple (π,ψ,τ) where π is an irreduciblecuspidal automorphic representation of GL2 and ψ and τ are Hecke characters of A×K such that ψ|A×Q = χπ .Then πψ τ is an automorphic representation of MP. We extend this to a representation of P by requiringthat NP act trivially. Then Klingen Eisenstein series are forms on GU(2,2) which are induced the aboverepresentation of P. In fact we need to first work locally for each place v (say, finite) of Q. Let (πv,ψv,τv)be the local triple then (πψ)v τv is a representation of MP(Qv). We extend it to a representation ρv ofP(Qv) by requiring that NP(Qv) acts trivially. Then we form the induced representation I(ρv) = IndG(Qv)

P(Qv)ρv.

When everything is unramified and φv is a spherical vector of πv, there is a unique vector f 0φv∈ I(ρv) which

is invariant under G(Zv) and f 0φv(1) = φv. The Archimedean picture is slightly different (see [24, section

9.1]).

Let φ = ⊗vφv ∈ π and let I(ρ) be the restricted product of the I(ρv)’s with respect to the unramified vec-tors above. If f ∈ I(ρ) we let fz(g) = δ (m)

32+zρ(m) f (k) for g = mnk ∈MPNPK. Here we let K be G(Z).

Note that the fz takes values in the representation space V of π . However π can be embedded to the spaceof automorphic forms on GL2(AQ). We also write fz(g) for the function on GU(2,2)(AQ) given by fz(g)(1).


The Klingen Eisenstein Series is defined by:

E( f ;z,g) := ∑γ∈P(Q)\G(Q)

fz(γg).

This is absolutely convergent for Rez >> 0 and can be meromorphically continued to all z ∈ C.

4.2 p-adic Families

Let I be a normal domain finite over Zp[[W ]]. (W is a variable) and f is a normalized ordinary eigenformwith coefficient ring I. In section 8 we are going to define the “Eisenstein Datum” D which contains theinformation of f,I,K, etc. Define ΛD := I[[ΓK]][[Γ−K ]]. We are going to define the set of arithmetic pointsφ ∈ SpecΛD and this ΛD p-adically parameterizes triples ( fφ ,ψφ ,τφ ) to which we associate the KlingenEisenstein series. Later we will also give ΛD the structure of a finite Λ2-algebra and construct a ΛD-adicnearly ordinary Klingen Eisenstein family, which we denote by ED.

Now we consider ΛD-adic cusp forms on GU(2,2). Let hD := hΣ ,0ord (K,ΛD) be the Hecke algebra for the

space of ΛD-coefficient nearly ordinary cuspidal forms with respect to some level group K. It is generatedby Hecke operators at primes outside Σ and the prime p.

Definition 9. Let ID be the ideal of hD generated by T−λED(T )’s for T elements in the abstract algebra.

Here λED(T ) is the Hecke eigenvalue of T on ED. The structure map ΛD → hD/ID is easily seen to be

surjective. Thus there is an ideal ED of ΛD such that ΛD/ED ' hD/ID. This ED is called the KlingenEisenstein ideal.

The motivation to define this ideal will be more clear after we have discussed the Galois representations.

5 Galois Representations and Lattice construction

5.1 Galois Representations

We first recall the following theorem (due to Harris-Taylor, S.W Shin, S. Morel, C.Skinner et al.) attachingGalois representations to automorphic representations on GU(n,n).

Theorem 4. Let π be an irreducible cuspidal representation of GU(n,n)(AQ) and let χπ be its centralcharacter. Let Σ(π) be the finite set of primes ` such that either π` or K is ramified. Suppose π∞ is theregular holomorphic discrete series of weight k := (kn+1, ...,k2n;k1, ...,kn) such that

k1 ≥ ...≥ kn,kn ≥ kn+1 +2n,kn+1 ≥ ...≥ k2n.

Then there is a continuous semisimple representation:

Rp(π) : GK→ GLn(Qp)

such that:

(i) Rp(π)∨(1−2n)⊗σ1+c

χπ' Rp(π)

c.(ii) Rp(π) is unramified outside primes above those in Σ(π)∪p and for such primes w we have

det(1−Rp(π)(frobw)q−sw ) = L(BC(π)w⊗ψw,s+

12−n)−1.

(iii) If π is nearly ordinary at p, then:

44 Xin Wan

Rp(π)GKv0'

ξ2n,v0ε−κ2n ∗ ∗0 ... ∗0 0 ξ1,v0ε−κ1

and

Rp(π)|GK,v0'

ξ1,v0εκ1+1−2n−|k| ∗ ∗0 ... ∗0 0 ξ2n,v0εκ2n+1−2n−|k|

.

Here ξi,v0 and ξi,v0 are unramified characters and ε is the cyclotomic character, |k| = k1 + ...+ k2n, κi =ki +n− i for 1≤ i≤ n and κi = ki +3n− i for n+1≤ i≤ 2n.

Returning to the GU(2,2) case, it is formal to patch the Galois representations attached to cuspidal nearlyordinary forms to a Galois pseudo-character RD of GK with values in hD. (Pseudo characters are firstlyintroduced by Wiles [14]. They are function on GK satisfying the relations that should be satisfied by thetrace of a representation. However it does not necessarily come from a representation. We omit the defini-tions.) We can associate a Galois representation ρED

to the Klingen Eisenstein family ED with coefficientring ΛD by a similar recipe. It is essentially the direct sum of the Galois representation ρf associated to theHida family f with two ΛD-adic characters.

The motivation for the Klingen Eisenstein ideal is:

RD(modID) = trρED(modED).

(Recall that hD/ID ' ΛD/ED).) This relation follows from the congruences for the corresponding Heckeeigenvalues. Also, RD is generically “more irreducible" than ρED

in the sense that it can be written as thesum of at most two “generically irreducible” pseudo-characters while ρED

has three pieces. (This is provenin [24, 7.3.1] using a result of M.Harris on non-existence of CAP forms of very regular weights.)

The next thing to do is use the “lattice construction" to get the Galois cohomology classes from the congru-ences between irreducible and reducible Galois representations.

Recall in the last section we have:ΛD/ED

∼−→ hD/ID

traceρED(modED) = RD(modID).

Our goal is to prove:(LΣ

1,QLΣf,K)⊃ ED ⊃ charΣ

f,K.

Now we are going to prove the second inclusion using the lattice construction. The first one will be provedat the end of section 9.

5.2 Galois Argument: Lattice Construction

The lattice construction in [24] involves 3 irreducible pieces and is complicated. Instead we are going togive the lattice construction which involves only 2 pieces (the case in [15]) and briefly mention the differ-ence at the end. Let us axiomize the situation: let Λ be the weight algebra and I a reduced ring which is afinite Λ -algebra. Let ρ be a Galois representation of GQ on I2. Let J and I be nonzero ideals of Λ and I suchthat the structure map induces Λ/J ' I/I. Let P be a height one prime of Λ such that ordP(J) = t > 0. Thenthere is a unique height one prime P′ of I containing (I,P). Since I is reduced we can talk about its totalfraction ring K =∏i FJi where the Ji’s are domains finite over I and the FJi ’s are the fraction fields of the Ji’s.

Suppose:

1 Each representation ρJi on F2Ji

induced from ρ via projection to FJi is irreducible.2 There are Λ×-valued characters χ1 and χ2 of GQ such that:


trρ(σ)≡ χ1(σ)+χ2(σ)(modI)

for each σ ∈ GQ.3 There are I×-valued characters χ ′1 and χ ′2 of GQp such that

ρ|GQp'(

χ ′1 ∗χ ′2

)and there is a σ0 ∈ GQp such that χ ′1(σ0) 6≡ χ ′2(σ0)(modP′).

4 χ1(σ)≡ χ ′1(σ)(mod I), χ2(σ)≡ χ ′2(σ)(mod I) for each σ ∈ I[GQp ].5 ρ is unramified outisde p.

We define the dual Selmer group X := H1ur(Q,Λ ∗(χ−1

1 χ2))∗. Here “ur” means extensions unramified every-

where and ∗ means Pontryagin dual.

Definition 10. Let charΛ (X) be the characteristic ideal of X as a Λ module.

We are going to prove:

Proposition 1. Under the assumptions above, ordP(charΛ (X))≥ ordP(J).

Proof. Suppose t = ordP(J) > 0. We take the σ0 in assumption (3) and a basis (v1,v2) so that ρ(σ0) has

the form(

χ1(σ0)χ2(σ0)

). We write ρ(σ) =

(aσ bσ

cσ dσ

)∈M2(K) for each σ ∈ K[GQ] with respect to this

basis. Then we claim the following. Let r := χ1(σ0)−χ2(σ0), (so r 6∈ P)

a raσ ,rdσ ,r2bσ cτ ∈ I for all σ ,τ ∈ I[GQ] and raσ ≡ rχ1(σ)(modI), rdσ ≡ rχ2(σ)(modI), r2bσ cτ ≡0(modI).

b C := cσ : σ ∈ I[GQ] is a finite faithful A-module;c cσ = 0 for σ ∈ Ip.

(c) is by (3) and (b) follows easily from assumption (1) and GQ being compact. (a) is by calculation: e.g.set δ1 := σ0−χ2(σ0) then raσ = traceρ(δ1σ)≡ rχ1(σ)(modI) by assumption (2).

Now we deduce the proposition using these claims. We write ΛP, IP, etc for the localizations at P anddefine M := IP[GQ]v1 ⊂ V . Then it is easy to check that M = IPv1⊕CPv2. Define M2 := CPv2. Then (a)implies M2 := M2/IM2 ⊂ M := M/IM is a direct summand that is GQ stable. Define M1 = IPv1 andM1 =M1/IM1 then we have

0→ M2→ M→ M1→ 0.

Now we return to the lattice construction without localization at P. We will find a finite Λ -module m2 ⊆ M2such that

i m2,P = M2;ii there exists a Λ -map X →m2 that is a surjection after localizing at P.

Then ordP(charΛ (X))= ordP(charΛP(XP))= ordP(FittΛP(ΛP/charΛP(XP)))≥ ordP(FittΛPm2,P)= ordP(FittΛPM2).But FittΛPM2(mod J) = FittΛP/JΛPM2 = FittIP/IIPM2 = FittIPM2(modI) = FittIPM2(modI) = 0, the lastequality follows from the fact that M2 is a faithful submodule of FracI in view of the generic irreducibilityof the Galois representation ρ . So FittΛPM2 ⊆ J and ordP(charΛ (X))≥ ordPJ.

Now let m ⊂ M be the I[GQ]-module generated by v1, m2 := m∩ M2, m1 := m/m2 ⊂ M1. Note thatm2,P = M2. Then we have:

0→m2→m→m1→ 0. (*)

• By assumption (5) and (c) above this extension is everywhere unramified.• M1 'Λ/PtΛ as Λ -module by definition. So it is easy to see m1 'Λ/PtΛ as well.• By (a) the GQ-action on m2 and m1 are given by χ2 and χ1 respectively.

We expect the (∗) in the matrix to give the desired extension. More precisely let [m]∈H1(Q,m2(χ−11 χ2))

be the class defined by (∗). Then we get a canonical map θ : HomΛ (m1,Λ∗)→ H1(Q,Λ ∗(χ−1

1 χ2)). Tak-ing the Pontryagin dual θ ∗ : H1(Q,Λ ∗(χ−1

1 χ2))∗→ m2. We claim that θ ∗ becomes surjective after taking

46 Xin Wan

localization at P. (As in Section 2 this is basically because m is generated by v1 over I[GQ].)

Proof of the claimLet R= ker(θ) and let S⊂R be any finite subset, mS := ∩φ∈Skerφ . Then we have:

0→m2/mS→∏φ∈S

Λ∗→∏

φ

Λ∗/(m2/mS)→ 0. (**)

Equip each module with the GQ action χ−11 χ2 and take the cohomology long exact sequence. By the defini-

tion of R the image of [m] in H1(Q,m2/mS(χ−11 χ2)) is in the kernel of the map H1(Q,m2/mS(χ

−11 χ2))→

H1(Q,∏φ∈S Λ ∗(χ−11 χ2)) which is a quotient of ∏φ∈S Λ ∗(χ−1

1 χ2)GQ which is killed by r = χ1(σ0)−

χ2(σ0) 6∈ P. Thus the exact sequence

0→ (m2/mS)P→ (m/mS)P→m1,P→ 0

is split. If (m2/mS)P 6= 0 then this contradicts the fact that m is generated by v1 over I[GQ]. Thus m2,P =mS,P. By the arbitrariness of S we get RP = 0. This proves the claim.

Now we compare with the [24] case. There we have 3 irreducible pieces and the matrix is like

χ1∗ ρ f∗ χ2

.

We expect the upper ∗ in the matrix to give the required extension. However we are not able to distinguishthe contribution of (∗) to H1

f (K,χ−11 ρ f ) and H1

f (K,χ−11 χ2)

c=1 = H1f (Q,τ) where τ is the composition

of the transfer map V : GabQ → Gab

K and χ−11 χ2. But by the Iwasawa main conjecture for Hecke characters

proved in [15], this H1f (K,χ−1χ2)

c=1 is controlled by the p-adic L-function for the trivial character, whichis a unit.

6 Doubling Methods

6.1 Siegel Eisenstein Series on GU(n,n)

Let Qn be the Siegel parabolic consists of block matrices(× ××

). Let v be a finite prime of Q, write

Kn,v for GU(n,n)(Zv). Fix χ a character of K×v . Let In(χ) be the space of smooth and Kn-finite functions

f : Kn,v→ C such that f (qk) = χ(detDq) f (k) for q =

(Aq Bq

Dq

)∈ Qn from such f we define

f (z,−) : Gn(Qv)→ C

byf (z,qk) := χ(detDq)|detAqD−1

q |z+ n

2v f (k).

Suppose Kv is unramified over Qv and χ is unramified, then there is a unique vector f 0 ∈ I(χ) which isinvariant under Kn,v and f 0(1) = 1. There is an Archimedean picture as well (See [24, 11.4.1]).

Now let χ = ⊗vχv be a Hecke character of A×K/K×. Then we define I(χ) as a restricted product of lo-cal I(χv)’s as above with respect to the above unramified vectors. For any f ∈ I(χ) we define the SiegelEisenstein series

E( f ;z,g) := ∑γ∈Qn(Q)\Gn(Q)

f (z,γg).

This is absolutely convergent if Rez >> 0 and has a meromorphic continuation to all z ∈ C.


6.2 Some Embedding

The Klingen Eisenstein series are difficult to compute, while Siegel Eisenstein series are much easier. Thepoint of doubling method is to reduce the computation of the former to the latter. We are going to introducesome important embeddings that are used in the Pullback formulas. Let (V1,ω1) be the Hermitian spacefor U(1,1) and (V−1 ) another Hermitian space whose metric is (−ω1). Elements of V1 and V−1 are denoted(v1,v2) and (u1,u2) for vi,ui ∈ K. Let V2 = V1⊕X ⊕Y be the Hermitian space for U(2,2) where X ⊕Y

is a 2-dimensional Hermitian space for the metric(

1−1

)and elements are written as (x,y) for x,y ∈ K

with respect to this basis. Let W =V2⊕V−1 be the Hermitian space for U(3,3). Let R =

1

11

11

1

and

S =

1

11−1 1

1−1 1

These give maps:

(v1,x,v2,y,u1,u2) 7→ (v1,x,u2,v2,y,u1)

(v1,x,u2,v2,y,u1) 7→ (v1−u1,x,u2− v2,v2,y,u1).

Now we define the embedding:

γ3 : G2,1 := (g,g′) ∈ GU(2,2,)×GU(1,1),µ(g) = µ(g′) → GU(3,3)

by (g

g′

)7→ S−1R−1

(g

g′

)RS.

Let V d be the image of V1 in V1⊕V−1 by the diagonal embedding. Let τ1 be any element of U(3,3)(Q)

which maps the maximal isotrophic subspace V d⊕X to Kv1⊕Ku1⊕X , then one can check that: τ−11 Q3τ1∩

γ3(U(2,2)×U(1,1)) = γ3(Q2×B1). An important property of such embedding is:

(m(g,x)n,g) : g ∈ GU(1,1),x ∈ ResK/QGm,n ∈ NP ⊂ Q3.

6.3 Pullback Formula

Let χ be a unitary Hecke character as before and f ∈ I(χ). Given a cusp form ϕ on G1, define the pullbacksection by:

Fϕ( f ,z,g) :=∫

U(1,1)(A)f (z,γ3(g,g1h))χ(detg1h)ϕ(g1h)dg1

where h ∈ GU(1,1)(A) is any element such that µ(h) = µ(g). This is absolutely convergent if Rez >> 0.It is easy to see that Fφ does not depend on the choice of h. Note that this is a Klingen Eisenstein section.Then

Proposition 2. For z in the region of absolute convergence and h as above, we have:∫U(1,1)(Q)\U(1,1)(A)

E( f ;z,γ3(g,g1h))χ(detg1h)ϕ(g1,h)dg1 = ∑P(Q)\GU(2,2)(Q)

Fϕ( f ;z,γg).

48 Xin Wan

Remark 5. The right hand side is nothing but the expression of the Klingen Eisenstein series.

Proof. This is proven by Shimura [9]. There Shimura proved the following double coset decomposision in(2.4) and (2.7) in loc.cit:

U(3,3) = Q3γ((U(2,2)×U(1,1))∪Q3τ1γ3(U(2,2)×U(1,1))

andQ3γ3(U(2,2)×U(1,1)) = ∪β∈U(2,2),ξ∈U(1,1)Q3γ3((β ,ξ )),

Q3τ1γ3(U(2,2)×U(1,1)) = ∪β∈Q2\U(2,2),γ∈B1\U(1,1)Q3τ1γ3((β ,γ)).

Thus by unfolding the Siegel Eisenstein series we write the integration into two parts. We claim that theintegration for the part involving terms with τ1 is 0. We first fix β and sum over the γ’s, this equals∫

B1(Q)\U(1,1)(A)f (z;τ1γ3((βg,g1h)))ϕ(g1h)dg1.

Recall we have noted that τ1γ3(1,B1)τ−11 ⊆ Q3. Since ϕ is cuspidal,

∫B1(Q\B1(AQ)

φ(bg1h)db = 0 for allg1. Thus the integration is 0. This proves the claim. The proposition then follows from our description forQ3γ3(U(2,2)×U(1,1)).

7 Constant Terms

Suppose φ is of weight κ and let zκ = κ−32 . Let P be the Klingen parabolic and R any standard Q parabolic of

GU(2,2). We are going to compute the constant terms E( f ,z,g)R of the Klingen Eisenstein series E( f ,z,g)along R. We write NR for the unipotent radical of P. The constant term along R is given by:

E( f ,z,g)R =∫

NR(Q)\NR(A)E( f ,z,ng)dn.

A famous computation of Langlands tells us that: if R 6= P then E( f ,z,g)R = 0. For R = P we first definethe intertwining operator:

A(ρ,z, f )(g) :=∫

NP(A)fz(wng)dn.

This is absolutely convergent for Rez >> 0 and is defined for all z ∈ C by meromorphic continuation. Itis a product of local integrals. This intertwins the representations I(ρ) and some I(ρ1) where ρ1 is definedsimilar to ρ but replacing (π,ψ,τ) by (π× (τ det),ψττc, τc). Then E( f ,z,g)P = fz(g)+A(ρ,z, f )(g). Itturns out that under our choices z = zκ and κ > 6, A(ρ,z, f ) is absolutely convergent and the Archimedeancomponent is 0. Thus A(ρ,zκ , f ) equals 0. Thus

E( f ,zκ ,g)P = fzκ(g).

Let us explain how the special L-values that we are interested in show up in the constant term of the KlingenEisenstein series. The Klingen section f is realized as the pullback section of some Siegel Eisenstein serieson GU(3,3). At the unramified places a computation of Lapid and Rallis [5] tells us that if the Siegel sectionis f 0

v then the pullback section is f 0φv

L(π,ψ/τ,z+1)L(χπ(ψ/τ)′,2z+1) Here the first L-factor is the localEuler factor for the base change of the dual π twisted by ψ/τ and the second is a Dirichlet L-factor. Takingthe product over all good primes, the special L-values we are interested in show up as the constant term ofthe Klingen Eisenstein series obtained by pullback.

8 p-adic Interpolation

Definition 11. An Eisenstein datum is a sextuple D := (A,I, f,ψ,ξ ,Σ) where


• A is a finite Zp-algebra and I is a normal domain finite over A[[W ]].• f is a Hida family of cuspidal newforms with coefficient ring I.• ψ is an A-valued finite order character which restricts to the tame part of the central character of f on A×Q.• ξ is another A-valued finite order Hecke character of A×K.• Σ is a finite set of primes containing all the bad primes.

Recall that we have defined a ring ΛD = I[[ΓK]][[Γ−K ]]. We use ΛD to interpolate triples ( f ,ψ,τ) that areused to construct Klingen Eisenstein series. Recall that we have defined a weight ring Λ2 ' Zp[[Γ2]] forΓ2 ' (1+ pZp)

4. We first give ΛD a Λ2-algebra structure. We define homomorphisms α : A[[ΓK]]→ I[[Γ−K ]]and β : A[[ΓK]]→ I[[ΓK]] (we omit the formulas). Then the Λ2-algebra structure map is given by composingα⊗β : A[[ΓK×ΓK]]→ΛD with the map Γ2→ ΓK×ΓK given by:

(t1, t2, t3, t4) 7→ recK(t3t4, t−11 t−1

2 )× recK(t4, t−12 ),

where recK is the reciprocity map in class field theory normalized by the geometric Frobenius. Let ψψψ :=α ω−1ψΨ

−1K and ξξξ := β χfξΨK.

Definition 12. A point φ ∈ SpecΛD is called arithmetic if φ |I is arithmetic with some weight κφ ≥ 2 andthere are ζ±,ζ

′− ∈ µp∞ such that φ(γ+) = ζ+(1+ p)κφ−2,φ(γ−) = ζ− for γ± ∈Γ

±K and φ(γ−

′) = ζ ′− for γ−

′

the topological generator of Γ−K .

For every such φ we define Hecke characters. Let p = v0v0 be the decomposition in K and let

ψφ (x) := x−κφ

∞ xκφ

v0 (φ ψψψ)(x), ξφ = φ ξ .

Then we can construct a ΛD-coefficient formal q-expansion Esieg that, when specialize to a Zariski denseset of arithmetic points φ , is the nearly ordinary Klingen Eisenstein series EKling,φ we constructed using the

triple: (fφ , ψφ |.|κφ

2 ,τφ = ψφ ξ−1φ|.|

κφ

2 ). This is achieved by first constructing a ΛD-adic Siegel Eisensteinseries on GU(3,3) and using the pullback formula to construct the Klingen Eisenstein family on GU(2,2).To do this we need to choose a Siegel section fφ at each arithmetic point φ so that

1 fφ depends p-adic analytically on φ ;2 the pulls back of fφ to (a multiple of) the nearly ordinary Klingen Eisenstein section.

The hardest part is the computations at the primes dividing p ([24, 11.4.14,15,19]). It turns out that certainSiegel-Weil Eisenstein sections work well. In fact in [24], the section is not given in terms of the Siegel-Weilsection. However it indeed provided the idea of how the section given in loc.cit is figured out. Let us brieflyexplain the idea.

Let Φ be the Schwartz function on M3,6)(Qp) defined by:

Φ(X ,Y ) := Φ1(X)Φ2(Y ),

where X and Y are 3×3 matrices and define a Siegel-Weil section by:

f Φ(g) = χ−12 (detg)|detg|−s+ 3

2p ×

∫GL3(Qp)

Φ((0,X)g)χ−11 χ

−12 (detX)|detX |−2s+3

p d×X

for χp = (χ1,χ2). The Φ2 means the Fourier transform of Φ2. We let Φ1 be a Schwartz function supportedon the set of matrices X such that the X13 and X31 are in Z×p and the values on it is given by the product oftwo characters of X13 and X31. Choosing Φ2 properly and unfolding the formula for the β -th local Fouriercoefficients, we can make sure that it is essentially given by Φ1(

tβ ) (up to some easier constant dependingon β ). Thus the first requirement is ensured. This Siegel Weil section is explicitly given by

fp(g) = ∑a∈(Op/x)×

f 0,(2)z (g

(a−1

a

))

where (x) = cond(ξ c), τ is our χ defining the Siegel Eisenstein section, ξ = ψ/τ and f 0,(2)z in loc.cit lemma

11.4.20.

50 Xin Wan

How to interpolate the Klingen Eisenstein series?

Hida proved the existence of a Hecke operator 1 f ∈ Tordκ (N,χ f ,A)⊗A FA on the space Sord

κ (N,χ f ) of ordi-nary cusp forms with weight κ level N and character χ f . such that

1 f .g =

< g, f c|κ(−1

N

)>

< f , f c|κ(−1

N

)>

f .

This 1 f is not necessarily p-adically integral ([3]). The congruence number η f is defined (up to a p-adicunit) to be the minimally divisible by p number such that ` f := η f 1 f is in Tord

κ (N,χ f ,A). The candidatethat we choose for the Klingen Eisenstein series EKling,φ at the arithmetic point φ is the one such that:

`U(1,1)f eU(1,1)Esieg,φ |U(2,2)×U(1,1) = EKling,φ f .

Here the superscript means the Hecke operators are applied to the forms considered as a form on U(1,1).If we replace f by a Hida family f and suppose the local Hecke algebra Tm f (the localization of the Heckealgebra at the maximal ideal m f corresponding to f ) is Gorenstein, then we can similarly define 1f and ηf,`f, thus interpolating everything in p-adic families.

In particular, we get the Klingen-Eisenstein series interpolating EKling,φ whose constant terms are divisibleby LΣ

f,K.LΣ1 in light of the discussion at the end of the last section.

9 Fourier-Jacobi Coefficients

Recall that we have seen that the constant terms of the Klingen Eisenstein family are divisible by the p-adicL-function. In order to show that the Eisenstein ideal is contained in the principal ideal (LΣ

f,K), we still needto show that some Fourier coefficient is co-prime to the p-adic L-function.

9.1 Generalities

We are going to compute the Fourier-Jacobi coefficient of the Siegel Eisenstein serie Esieg as a function onU(1,1)(A) via the embedding γ3 : U(2,2)×U(1,1) →U(3,3). The purpose is, by the pullback formulaintroduced in the previous section, to express the Fourier coefficients of the Klingen Eisenstein series interms of the Petersson inner product with the cusp form we start with. For Z ∈H3

Esieg(Z) = ∑T≥0

aT e(TrTZ).

Write S2(Q) or S2(Qv) for the set of 2×2 Hermitian matrices over Q or Qv. For β a 2×2 Hermitian matrixthe β th Fourier-Jacobi coefficient is

∑

T=

(β ∗∗ ∗

)aT e(TrTZ).

We have an integral representation for the Fourier-Jacobi coefficients:

Esieg,β (g) =∫

NQ(Q)\NQ(A)Esieg(

13S 00 013

g)eA(−TrβS)dS.


Here eA = ⊗ev and ev(xv) = e−2πixv for v a finite primes and ev(xv) = e2πxv for x ∈ ∞. The followinglemma gives a way to compute the Fourier-Jacobi coefficients of Esieg.

Lemma 1. Let f ∈ I3(χ), β ∈ S2(Q). Suppose β > 0. Let V be the 2-dimensional K-space of columnvectors. If Re(z)> 3

2 . Then:

Esieg,β ( f ;z,g) = ∑γ∈Q1(Q)\G1(Q)

∑x∈V

∫S2(A)

f (w3

13S xtx 013

α1(1,γ)g)eA(−TrβS)dS.

Proof. We omit it here. See [24, 11.3]

The integrand in the lemma is a product of local integrals. We are mainly interested in evaluating the FourierJacobi coefficients at α1(diag(y, ty−1),g)) for y ∈ GL2(AK) and g ∈U1(AQ).

Definition 13. For each prime v of Q and f ∈ I3(χv), set

FJβ ( f ;z,x,g,y) =∫

S2(Qv)f (z,w3

13S xtx 013

α1(diag(y, ty−1),g))ev(−TrβS)dS.

We are going to identify the Fourier Jacobi coefficients with some forms that we are more familiar with.

9.2 Backgrounds for Theta Functions

Local PictureLet v be a prime of Q and h ∈ S2(Qv), deth 6= 0. Then h defines a two-dimensional Hermitian space Vv. LetUh be the corresponding unitary group. Let λv be a character of K×v whose restriction to Q×v is trivial. Onecan define the Weil representation ωh,λ of Uh(Qv)×U(1,1)(Qv) on the space S(Vv) of Schwartz functionson Vv (we omit the formulas).

Global PictureNow let h ∈ S2(Q),h > 0 and a Hecke character λ =⊗λv of A×K/K× such that λ |A×Q = 1. Then we define

a Weil representation ωh,λ of Uh(AQ)×U(1,1)(AQ) on S(V ⊗A) by tensoring the local representations.

Theta FunctionsGiven Φ ∈ S(V ⊗AQ) we define

Θh(u,g;Φ) := ∑x∈V

ωh,λ (u,g)Φ(x)

which is an automorphic form on Uh(AQ)×U1(AQ) and gives the theta correspondence between Uh andU(1,1).

9.3 Coprime to the p-adic L-function

Now let us return to the Fourier Jacobi coefficients. It turns out that by some local computations, for eachv, FJβ ( f ;z,x,g,y) has the form f1(g)(ωβ ,λv(y,g)Φv)(0) where we have chosen a Hecke character λ asabove, f1 ∈ I1(χv/λv) and Φv is a Schwartz function on K2

v , ωβ ,λv is defined using the character λv. Thusfrom lemma 1 the Fourier Jacobi coefficient is the product of an Eisenstein series E1(g) and a theta seriesΘβ (y,g).

Now we prove that the Klingen Eisenstein series is coprime to the p-adic L-function. Let us take an auxil-iary Hida family g of cuspidal eigenforms. Using the functorial property of the theta correspondence we can

52 Xin Wan

find some linear combinations of Eβ ( f ;zκ ,α1(diag(y, ty−1),g))’s which “picks up" the g-eigencomponentof Θβ (y,g) (as a function of g). By pairing this with the original φ ∈ π we started with, we find certainlinear combinations of the Fourier coefficients of the Klingen Eisenstein family which can be expressed inthe form AgBg where Bg is the “multiple" of g showing up in Θβ (y,g). By choosing g properly Bg canbe made a unit in ΛD (g is chosen to be a Hida family of theta series from the quadratic imaginary fieldK. Bg interpolates a square of central critical values of Hecke L-functions of CM characters. One needs touse a result of Finis [2] on the non-vanishing modulo p of anticyclotomic Hecke L-values to conclude Bgcan be chosen to be a unit). The factor Ag is interpolating < E1(g).g, fφ >, essentially the Rankin SelbergL-values of g with f. By checking the nebentypus we find that Ag only involves I[[Γ +

K ]] and is non-zero bythe temperedness of f and g.

Now we make the following assumption: N = N+N− where N+ is a product of primes split in K and N− isa square-free product of an odd number of primes inert in K. Furthermore we assume that for each `|N−, ρfis ramified at `. Under this assumption Vatsal [12] proved if we expand the p-adic L-function as:

LΣf,K = a0 +a1(γ

−−1)+a2(γ−−1)2 + ......

for ai ∈ I[[Γ +K ]], then some ai must be in (I[[Γ +

K ]])×. This implies (easy exercise) that Ah is outside anyheight one prime P of I[[ΓK]] containing LΣ

f,K (since Ag belongs to I[[Γ +K ]], one may assume P = P+I[[Γ +

K ]]

for some height one prime P+ of I[[Γ +K ]]. By Vatsal’s result, ordP+(L

Σf,K) = 0.

We are ready to prove the result promised in the previous section: ordP(ED)≥ ordP(LΣf,K.LΣ

1 ) for any heightone prime P (Here LΣ

1 is the p-adic L-function for the trivial character, which is co-prime to (LΣf,K) by the

work of Vatsal. In [24] Skinner-Urban actually worked in a more general setting by allowing non-trivialcharacters). First recall that all constant terms of the Klingen Eisenstein family are divisible by LΣ

f,KLΣ1 .

By the fundamental exact sequence one can find some family F of forms on GU(2,2) such that ED −(LΣ

f,K)(LΣ1 )F := H is a cuspidal family. Now we prove the desired inequality. Suppose r = ordP(L

Σf,K)≥ 1.

By construction there is a Fourier coefficient of the above constructed cuspidal family H outside P. Denoteit as c(β ,x;H) where β ∈ S2(Q) and x ∈ GU(2,2)(AQ). We define a map:

µ := hD→ΛP/PrΛP

by: µ(h) = c(β ,x;hH)/c(β ,x;H). This is ΛD-linear and surjective. Moreover,

c(β ,x;hH)≡ c(β ,x;hED)≡ λD(h)c(β ,x;ED)≡ λD(h)c(β ,x,H)(mod Pr).

Thus ID ⊆ kerµ . So we have a surjection µ : hD/IDΛP/PrΛP. But the right hand side is ΛD/ED. Thisgives the inequality.

10 Generalizations of the Skinner-Urban Work

We have seen that the key ingredient of this work is a study of the p-adic properties of the Fourier coefficientsof the Klingen Eisenstein series. To generalize this argument to more general unitary groups we need somenon-vanishing modulo p results for special values of L-functions, which so far is only available for forms onunitary groups of rank at most 2. We are able to study the Klingen Eisenstein series for U(1,1) →U(2,2)and U(2,0) →U(3,1), proving the corresponding main conjectures for two different Rankin Selberg p-adicL-functions. Here we only mention the following by product (proved in the author thesis):

Theorem 5. Let F be a totally real field in which p splits completely. Let f be a Hilbert modular form overF with trivial character and parallel weight 2. Let ρ f be the p-adic representation of GF associated to f .Suppose:

1 f has good ordinary reduction at all primes dividing p;2 ρ f is absolutely irreducible.

If the central value L( f ,1) = 0, then H1f (F,ρ f )is infinite.


In the case when the sign of L( f ,s) is −1 this is an early result of Nekovar [7] and Zhang [16]. The caseswhen this sign is +1 is new. Note that even in the case when F =Q our result is slightly stronger than theone in [24]. The reason is that by working with general totally real fields we can use a base change trick toremove some of the technical local conditions [24].

References

1. Y. Amice and J. Velu, Distributions p-adiques associes aux sries de Hecke, Asterisque, Nos. 24-25, Soc. Math. France,Paris, pp. 119-131, 1975.

2. T. Finis, Divisibility of anticyclotomic L-functions and theta functions with complex multiplication, Ann. of Math. (2)163 (2006), no. 3, 767-807.

3. H. Hida, A p-adic measure attached to the zeta functions associated with two elliptic modular forms. I, Invent. Math. 79(1985), no. 1, 159-195.

4. K. Kato, p-adic Hodge theory and values of zeta functions of modular forms, Cohomologies p-adiques et applicationsarithmetiques. III. Asterisque No. 295 (2004), ix, p. 117-290.

5. E. Lapid and S. Rallis, On the local factors of representations of classical groups in Automorphic representations, L-functions and applications: progress and prospects, 309-359, Ohio State Univ. Math. Res. Inst. Publ. 11, de Gruyter,Berlin, 2005.

6. B. Mazur, J. Tate, and J. Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math.84 (1986), no. 1, 1-48.

7. J. Nekovar, Selmer complexes, Asterisque 310 (2006).8. Ribet, Kenneth A., A modular construction of unramified p-extensions of Q(ζp), Invent. Math. 34 (1976), no. 3, 151-162.9. G. Shimura, Euler products and Eisenstein series, CBMS Regional Conference Series in Mathematics 93, American

Mathematical Society, Providence, RI, 1997.10. C.Skinner: Galois representations, Iwasawa theory and special values of L-functions, CMI Lecture Notes, 2009.11. C.Skinner and E.Urban: The Iwasawa Main Conjecture for GL2, 2010.12. V. Vatsal, Special values of anticyclotomic L-functions. Duke Math. J. 116 (2003), no. 2, 219-261.13. X. Wan, Iwasawa Theory for Unitary Groups, Thesis, 2012.14. A. Wiles, On ordinary Λ -adic representations associated to modular forms, Inv. Math. 94 (1988), 529-573.15. A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. 131, 1990.16. S. Zhang, Heights of Heegner points on Shimura curves. Annals of Mathematics, 153 (2001), 27-147

On extra zeros of p-adic L-functions:the crystalline case

Denis Benois

Abstract We formulate a conjecture about extra zeros of p-adic L-functions at near central points whichgeneralizes the conjecture formulated in [3] to include non-critical values. We prove that this conjecture iscompatible with Perrin-Riou’s theory of p-adic L-functions. Namely, using Nekovár’s machinery of Selmercomplexes we prove that our L -invariant appears as an additional factor in the Bloch–Kato type formulafor special values of Perrin-Riou’s module of L-functions.

Nous avons toutefois supposé pour simplifier queles opérateurs 1−ϕ et 1− p−1ϕ−1 sontinversibles laissant les autres cas, pourtantextrêmement intéressants pour plus tard.

Introduction to Chapter III of [55]

Introduction

0.1 Extra zeros

Let M be a pure motive over Q. Assume that the complex L-function L(M,s) of M extends to a meromorphicfunction on the whole complex plane C. Fix an odd prime p. It is expected that one can construct p-adic analogues of L(M,s) p-adically interpolating algebraic parts of its special values. The above programhas been realised and the corresponding p-adic L-functions constructed in many cases, but the generaltheory remains conjectural. In [55], Perrin-Riou formulated precise conjectures about the existence andarithmetic properties of p-adic L-functions in the case where the p-adic realisation V of M is crystallineat p. Let Dcris(V ) denote the filtered Dieudonné module associated to V by the theory of Fontaine. Let Dbe a subspace of Dcris(V ) of dimension d+(V ) = dimQp V c=1 stable under the action of ϕ . (As usual, cdenotes the complex conjugation.) One says that D is regular if one can associate to D a p-adic analogue ofthe six-term exact sequence of Fontaine and Perrin-Riou (see Sect. 3.1.3 and [55] for an exact definition).Fix a lattice T of V stable under the action of the Galois group and a lattice N of a regular module D.Perrin-Riou conjectured that one can associate to this data a p-adic L-function Lp(T,N,s) satisfying someexplicit interpolation property. Let r denote the order of vanishing of L(M,s) at s = 0 and let L∗(M,0) =lims→0 s−rL(M,s). Then at s = 0 the interpolation property reads

lims→0

Lp(T,N,s)sr = E(V,D)RV,D(ωV,N)

L∗(M,0)RM,∞(ωM)

.

Denis BenoisInstitut de Mathématiques, Université de Bordeaux 351, cours de la Libération 33405 Talence, France, e-mail: [email protected]

55

[email protected]

[email protected]

56 Denis Benois

Here RM,∞(ωM) (resp. RV,D(ωV,N)) is the determinant of the Beilinson (resp. the p-adic) regulator computedin some compatible bases ωM and ωV,N and E(V,D) is an Euler-like factor given by

E(V,D) = det(1− p−1ϕ−1 | D) det(1−ϕ | Dcris(V )/D).

If either Dϕ=p−1 6= 0 or (Dcris(V )/D)ϕ=1 6= 0 we have E(V,D) = 0 and the order of vanishing of Lp(N,T,s)should be > r. In this case we say that L(T,N,s) has an extra zero at s = 0. The same phenomenon occursin the case where V is semistable and non-crystalline at p. An archetypical example is provided by ellipticcurves having split multiplicative reduction [18]. Assume that 0 is a critical point for L(M,s) and thatH0(Q,V ) = H0(Q,V ) = 01. In [3] using the theory of (ϕ,Γ )-modules we associated to each regular Dan invariant L (V,D) ∈ Qp generalising both Greenberg’s L -invariant [Gre00] and Fontaine–Mazur’s L -invariant [Mar11]. This allows one to formulate a quite general conjecture about the behavior of p-adicL-functions at extra zeros in the spirit of [Gre00]. To the best of our knowledge this conjecture is actuallyproved in the following cases:

1) Kubota–Leopoldt p-adic L-functions [32], [36]. Here the L -invariant can be interpreted in terms ofGross’ p-adic regulator [34]. We also remark that in [22] this result was generalized to totally real fields(assuming the Leopoldt conjecture).

2) Modular forms of even weight [Kak13], [34], [25]. Here the L -invariant coincides with Fontaine–Mazur’s L ( f ). In [17], [63] and [61] the method of [34] was generalized to study trivial zeros of ellipticcurves/modular forms of weight 2 over totally real fields.

3) Modular forms of odd weight [4]. The associated p-adic representation V is either crystalline orpotentially crystalline at p and we do need the theory of (ϕ,Γ )-modules to define the L -invariant.

4) Symmetric squares of modular forms having either split multiplicative or good reduction [62]. In thesplit multiplicative and good ordinary reduction cases the associated p-adic representation V is ordinary andthe L -invariant reduces to Greenberg’s construction [Gre00]. In the supersingular case again the definitionof the L -invariant involves (ϕ,Γ )-modules. For elliptic curves having good ordinary reduction anotherproof, based on factoring the several-variable Rankin–Selberg p-adic L-function along the diagonal hasbeen suggested by Dasgupta (work in progress). The fact that a factorization would lead to the proof of theconjecture appears previously in [16].

5) Symmetric powers of CM-modular forms [37], [39].

0.2 Extra zero conjecture

The goal of this paper is to generalise the conjecture from [3] to the non critical point case. Assume thatV is crystalline at p. Then a weight argument shows that E(V,D) can vanish only if wt(M) = 0 or −2. Inparticular, we expect that the interpolation factor does not vanish at s = 0 if wt(M) =−1 i.e. that the p-adicL-function can not have an extra zero at the central point in the good reduction case. To fix ideas assume thatwt(M) 6 −2 and that M has no subquotients isomorphic to Q(1) 2. We denote by H1

f (V ) the Bloch–KatoSelmer group of V. Then D is regular if and only if the associated p-adic regulator map

rV,D : H1f (V )→ Dcris(V )/(Fil0Dcris(V )+D)

is an isomorphism. The semisimplicity of ϕ : Dcris(V )→ Dcris(V ) (which conjecturally always holds) al-lows one to decompose D into a direct sum

D = D−1⊕Dϕ=p−1.

Under some mild assumptions (see Sect. 3.1.2 and 4.1.2 below) we associate to D an L -invariant L (V,D)which is a direct generalization of the main construction of [3]. The Beilinson–Deligne conjecture predictsthat L(M,s) does not vanish at s = 0 and that L(M∗(1),s) has a zero of order r = dimQp H1

f (V ) at s = 0.Let L∗p(T,N,0) denote the first non-zero coefficient in the Taylor expansion of Lp(T,N,s). We propose thefollowing conjecture:

1 By Tate’s conjecture this condition should be equivalent to the vanishing of H0(M) and H0(M∗(1)).2 The last condition is not really essential and can be suppressed.

On extra zeros of p-adic L-functions: the crystalline case 57

Conjecture 1. Let D be a regular subspace of Dcris(V ) and let e = dimQp(Dϕ=p−1

). Theni) The p-adic L-function Lp(T,N,s) has a zero of order e at s = 0 and

L∗p(T,N,0)RV,D(ωV,N)

= −L (V,D)E+(V,D)L(M,0)

RM,∞(ωM).

ii) Let D⊥ denote the orthogonal complement to D under the canonical dualityDcris(V )×Dcris(V ∗(1))→ Qp. The p-adic L-function Lp(T ∗(1),N⊥,s) has a zero of order e + r wherer = dimQ H1

f (V ) at s = 0 and

L∗p(T∗(1),N⊥,0)

RV ∗(1),D⊥(ωV ∗(1),N⊥)= L (V,D)E+(V ∗(1),D⊥)

L∗(M∗(1),0)RM∗(1),∞(ωM∗(1))

.

In the both cases

E+(V,D) = E+(V ∗(1),D⊥) =

= det(1− p−1ϕ−1 | D−1) det(1− p−1

ϕ−1 | Dcris(V ∗(1))).

Remarks 1) E+(V,D) is obtained from E(V,D) by excluding the zero factors. It can also be written in theform

E+(V,D) = E∗p(V,1)detQp

(1− p−1ϕ−1

1−ϕ|D−1

)where Ep(V, t) = det(1−ϕt | Dcris(V )) is the Euler factor at p and

E∗p(V, t) = Ep(V, t)(

1− tp

)−e

.

2) Assume that H1f (V ) = 0. Since H1

f (V∗(1)) should also vanish by the weight argument, our conjecture

in this case reduces to Conjecture 2.3.2 from [3].3) The regularity of D supposes that the localisation H1

f (V )→ H1f (Qp,V ) is injective. Jannsen’s conjec-

ture (made more precise by Bloch and Kato) says that the p-adic realisation map H1f (M)⊗Qp → H1

f (V )

is an isomorphism. The composition H1f (M)→ H1

f (Qp,V ) of these two maps is essentially the syntomicregulator. Its injectivity seems to be a difficult open problem.

4) Let f =∞

∑n=1

an( f )qn and g =∞

∑n=1

bn( f )qn be two different newforms of weight 2 and nebentypus η f

and ηg on Γ1(N). For any prime l we denote by αl,1 and αl,2 (resp. βl,1 and βl,2) the roots of the Heckepolynomial X2−al( f )X +η f (l)l (resp. X2−bl( f )X +ηg(l)l). The Rankin–Selberg L-function L( f ⊗g,s)is defined by

L( f ⊗g,s) = ∏l

Pl( f ⊗g, l−s)−1

wherePl( f ⊗g,X) = (1−αl,1βl,1X)(1−αl,1βl,2X)(1−αl,2βl,1X)(1−αl,2βl,2X).

Let Wf and Wg be the p-adic representations associated to f and g respectively. Let Wf ,g = Wf ⊗Wg andTf ,g a fixed lattice of Wf ,g. The complex L-function L(Wf ,g,s) associated to Wf ,g coincides with L( f ⊗g,s)up to Euler factors at l | N. The p-adic representation Wf ,g(2) can be viewed as the p-adic realisation of amotive M f ,g(2) of weight −2. We remark that M f ,g(2) is non-critical and the special value L(Wf ,g(2),0) =L(Wf ,g,2) can be expressed in terms of the regulator map [1]. If p - N, the restriction of Wf ,g(2) on thedecomposition group at p is crystalline with Hodge–Tate weights (−2,−1,−1,0). The crystalline moduleDcris(Wf ,g(2)) is a 4-dimensional vector space generated by eigenvectors di j, 1≤ i, j≤ 2 such that ϕ(di j) =αiβ j p−2di j. Let Di j (16 i, j≤ 2) be the 3-dimensional subspace of Dcris(Wf ,g(2)) generated by dr,s, (r,s) 6=(i, j) and Ni j a fixed lattice of Di j. One expects that if f and g are not CM, then Di j is regular. If αp,rβp,s = p

for some (r,s) 6=(i, j), then Dϕ=p−1

i j 6= 0 and our extra zero conjecture predicts the behavior of Lp(Tf ,g,Ni j,s)

58 Denis Benois

at s = 0. As Wf ,g(2) is non-critical, this gives an example of extra zero which is not covered by the trivialzero conjecture from [3].

0.3 Selmer complexes and Perrin-Riou’s theory

In the last part of the paper we show that our extra zero conjecture is compatible with the Main Conjectureof Iwasawa theory as formulated in [55]. The main technical tool here is the descent theory for Selmercomplexes [23]. We hope that the approach to Perrin-Riou’s theory based on the formalism of Selmercomplexes can be of independent interest.

For a profinite group G and a continuous G-module X we denote by C•c (G,X) the standard complex ofcontinuous cochains. Let S be a finite set of primes containing p. Denote by GS the Galois group of themaximal algebraic extension of Q unramified outside S∪∞. Set RΓS(X) =C•c (GS,X) and RΓ (Qv,X) =C•c (Gv,X), where Gv is the absolute Galois group of Qv. Let Γ be the Galois group of the cyclotomic p-extension Q(ζp∞)/Q), Γ1 = Gal(Q(ζp∞)/Q(ζp)) and ∆ = Gal(Q(ζp)/Q). Let Λ(Γ ) = Zp[[Γ ]] denote theIwasawa algebra of Γ . Each Λ(Γ )-module X decomposes into the direct sum of its isotypical componentsX = ⊕

η∈∆X (η) and we denote by X (η0) the component which corresponds to the trivial character η0. Set

Λ = Λ(Γ )(η0). Let H denote the algebra of power series with coefficients in Qp which converge on theopen unit disk. We will denote again by H the associated large Iwasawa algebra H(Γ1) (see Sect. 2.2.1).In this paper we consider only the trivial character component of the module of p-adic L-functions becauseit is sufficient for applications to trivial zeros, but in the general case the construction is exactly the same.We keep notation and assumptions of Sect. 0.2. Assume that the weak Leopoldt conjecture holds for (V,η0)and (V ∗(1),η0). We consider global and local Iwasawa cohomology RΓIw,S(T ) = RΓS((Λ(Γ )⊗Zp T )ι) andRΓIw(Qv,T ) = RΓ (Qv,(Λ(Γ )⊗Zp T )ι) where ι is the canonical involution on Λ(Γ ). Let D be a regularsubmodule of Dcris(V ). For each non archimedean place v we define a local condition at v in the sense of[23] as follows. If v 6= p we use the unramified local condition which is defined by

RΓ(η0)

Iw (Qv,N,T ) = RΓ(η0)

Iw, f (Qv,T ) =[T Iv ⊗Λ

ι 1− fv−−−→ T Iv ⊗Λι

]where Iv is the inertia subgroup at v and fv is the geometric Frobenius. If v = p we define

RΓ(η0)

Iw (Qv,N,T ) = (N⊗Λ)[−1].

The derived version of the large exponential map ExpV,h, for h 0 (see [25]) gives a morphism

RΓ(η0)

Iw (Qp,N,T )→ RΓ(η0)

Iw (Qp,T )⊗H.

Therefore we have a diagram

RΓ(η0)

Iw,S (T )⊗Λ H //⊕v∈S

RΓ(η0)

Iw (Qv,T )⊗Λ H

(⊕v∈S

RΓ(η0)

Iw (Qv,N,T )

)⊗Λ H .

OO

Let RΓ(η0)

Iw,h (D,V ) denote the Selmer complex associated to this data. By definition it sits in the distinguishedtriangle


RΓ(η0)

Iw,h (D,V )→

(RΓ

(η0)Iw,S (V )

⊕(⊕v∈S

RΓ(η0)

Iw (Qv,D,V )

))⊗H→ (⊕

v∈S

RΓ(η0)

Iw (Qv,V )

)⊗H. (1)

Define

∆Iw,h(N,T ) =

det−1Λ

(RΓ

(η0)Iw,S (T )

⊕(⊕v∈S

RΓ(η0)

Iw (Qv,N,T )

))⊗detΛ

(⊕v∈S

RΓ(η0)

Iw (Qv,T )

).

Our results can be summarized as follows (see Theorems 4, 5 and Corollary 2).

Theorem 1. Assume that L (V,D) 6= 0. Theni) The cohomology RiΓ

(η0)Iw,h (D,V ) are H-torsion modules for all i.

ii) RiΓ(η0)

Iw,h (D,V ) = 0 for i 6= 2,3 and

R3Γ

(η0)Iw,h (D,V )'

(H0(Q(ζp∞),V ∗(1))∗

)(η0)⊗Λ H.

iii) The complex RΓ(η0)

Iw,h (D,V ) is semisimple in the sense that for each i the natural map

RiΓ

(η0)Iw,h (D,V )Γ → Ri

Γ(η0)

Iw,h (D,V )Γ

is an isomorphism.

This theorem allows us to apply to our Selmer complexes the descent machinery developed by Nekovárin [23]. Assume that L (V,D) 6= 0. Let K be the field of fractions of H. Then Theorem 1 together with (1)define an injective map

iV,Iw,h : ∆Iw,h(N,T )→K

and the module of p-adic L-functions is defined as

L(η0)Iw,h(N,T ) = iV,Iw,h(∆Iw,h(N,T ))⊂K.

Let γ1 be a fixed generator of Γ1. Choose a generator f (γ1−1) of the free Λ -module L(η0)Iw,h(N,T ) and define

a meromorphic p-adic functionLIw,h(T,N,s) = f (χ(γ1)

s−1),

where χ : Γ → Z∗p is the cyclotomic character. For a,b ∈Q∗p we will write a∼p b if a and b coincide up toa p-adic unit.

Theorem 2. Assume that L (V,D) 6= 0. Theni) The p-adic L-function LIw,h(T,N,s) has a zero of order e = dimQp(D

ϕ=p−1) at s = 0.

ii) One has

L∗Iw,h(T,N,0)

RV,D(ωT,N)∼p Γ (h)d+(V )L (V,D)E+(V,D)

#III(T ∗(1))Tam0ωM

(T )

#H0S (V/T )#H0

S (V∗(1)/T ∗(1))

,

where III(T ∗(1)) is the Tate–Shafarevich group of Bloch–Kato [4] and Tam0ωM

(T ) is the product of localTamagawa numbers of T .

Remarks 1) Using the compatibility of Perrin-Riou’s theory with the functional equation we obtain analo-gous results for the Lp(T ∗(1),N⊥,s) (see Sect. 5.2.6).

2) If Dcris(V )ϕ=1 = Dcris(V )ϕ=p−1= 0 the phenomenon of extra zeros does not appear, L (V,D) = 1 and

Theorem 2 was proved in [55], Theorem 3.6.5. We remark that even in this case our proof is different. We

60 Denis Benois

compare the leading term of L∗Iw,h(T,N,s) with the trivialisation iωM ,p : ∆EP(T )→Qp of the Euler–Poincaréline ∆EP(T ) (see [28]) and show that in compatible bases one has

L∗Iw,h(T,N,0)

RV,D(ωV,N)∼p Γ (h)d+(V )L (V,D)E+(V,D) iωM ,p (∆EP(T )) (2)

(see Theorem 5). Now Theorem 2 follows from the well known computation of iωM ,p (∆EP(T )) in terms ofthe Tate–Shafarevich group and Tamagawa numbers (see [31], Chapitre II).

3) Let E/Q be an elliptic curve having good reduction at p. Consider the p-adic representation V =

Sym2(Tp(E))⊗Qp, where Tp(E) is the p-adic Tate module of E. It is easy to see that D = Dcris(V )ϕ=p−1

is one dimensional. In this case some versions of Theorem 2 were proved in [56] and [Del89] with an adhoc definition of the L -invariant. Remark that p-adic L-functions attached to the symmetric square of anewform were constructed by Dabrowski and Delbourgo [24] and Rosso [62].

4) Another approach to Iwasawa theory in the non-ordinary case was developped by Pottharst in [58],[59]. Pottharst uses the formalism of Selmer complexes but works with local conditions coming from sub-modules of the (ϕ,Γ )-module associated to V rather than with the large exponential map. This approachhas many advantages, in particular it allows to develop an interesting theory for representations which arenot necessarily crystalline. Nevertheless it seems that the large exponential map is crucial for the study ofextra zeros at least in the good reduction case.

5) The Main conjecture of Iwasawa theory [55], [18] says that for h 0 the analytic p-adic L-functionLp(N,T,s) multiplied by a simple explicit Γ -factor ΓV,h(s) depending on h can be written in the formΓV,h(s)Lp(N,T,s) = f (χ(γ1)

s − 1) for an appropriate generator f (γ1 − 1) of L(η0)Iw,h(N,T ). Therefore the

Main conjecture implies Bloch–Kato style formulas for special values of Lp(N,T,s). We remark that theBloch–Kato conjecture predicts that

L∗(M,0)RM,∞(ωM)

∼p#III(T ∗(1))Tam0

ωM(T )

#H0S (V/T )#H0

S (V∗(1)/T ∗(1))

and therefore Theorem 2 implies the compatibility of our extra zero conjecture with the Main conjecture.Note that this also follows directly from (2) if we use the formalism of Fontaine and Perrin-Riou [28] toformulate Bloch-Kato conjectures.

0.4 The plan of the paper

The organisation of the paper is as follows. In Sect. 1 we review the theory of (ϕ,Γ )-modules which is themain technical tool in our definition of the L -invariant. We also give the derived version of the computationof Galois cohomology in terms of (ϕ,Γ )-modules. This follows easily from the results of Herr [40] and Liu[18] and the proofs are placed in Appendix. Similar results can be found in [58], [59]. In Sect. 2 we recallpreliminaries on the Bloch–Kato exponential map and review the construction of the large exponential mapof Perrin-Riou given by Berger [9] using again the basic language of derived categories. The L -invariantis constructed in Sect. 3.1. In Sect. 3.2 we relate this construction to the derivative of the large exponentialmap. This result plays a key role in the proof of Theorem 2. The extra zero conjecture is formulated inSect. 4. In Sect. 5 we interpret Perrin-Riou’s theory in terms of Selmer complexes and prove Theorems 1and 2.

This paper is a revised and extended version of the preprint [5]. In [5], Theorems 1 and 2 above wereproved under the additional assumption that H1

f (V ) = 0. As we pointed out before, in this case our extra-zero conjecture reduces to the Conjecture 2.3.2 from [3].

Acknowledgements I am very grateful to Jan Nekovár for interesting discussions and Daniel Delbourgo, Chung Pang Mokand Thong Nguyen Quang Do for useful remarks. I would especially like to thank the referee for numerous suggestions inimproving the paper and for the very patient correction of many minor points.


1 Preliminaries

1.1 (ϕ,Γ )-modules

1.1.1 The Robba ring (see [7],[19])

In this section K is a finite unramified extension of Qp with residue field kK , OK its ring of integers, and σ

the absolute Frobenius of K. Let K an algebraic closure of K, GK = Gal(K/K) and C the completion of K.

Let vp : C→ R∪∞ denote the p-adic valuation normalized so that vp(p) = 1 and set |x|p =(

1p

)vp(x).

Write B(r,1) for the p-adic annulus B(r,1) = x ∈C | r 6 |x| < 1. As usually, µpn denotes the group ofpn-th roots of unity. Fix a system of primitive roots of unity ε = (ζpn)n>0, ζpn ∈ µpn such that ζ

ppn = ζpn−1

for all n. Set Kn = K(ζpn), K∞ =⋃

∞n=0 Kn, HK = Gal(K/K∞), Γ = Gal(K∞/K) and denote by χ : Γ → Z∗p

the cyclotomic character.Set

E+ = lim←−x 7→xp

OC/ pOC = x = (x0,x1, . . . ,xn, . . .) | xpi = xi ∀i ∈ N.

Let xn ∈OC be a lifting of xn. Then for all m> 0 the sequence xpn

m+n converges to x(m) = limn→∞ xpn

m+n ∈OC

which does not depend on the choice of liftings. The ring E+ equipped with the valuation vE(x) = vp(x(0))is a complete local ring of characteristic p with residue field kK . Moreover it is integrally closed in its fieldof fractions E = Fr(E+).

Let A = W (E) be the ring of Witt vectors with coefficients in E. Denote by [ ] : E→W (E) the Teich-müller lift. Any u = (u0,u1, . . .) ∈ A can be written in the form

u =∞

∑n=0

[up−n]pn.

Set π = [ε]−1, A+K0

= OK0 [[π]] and denote by AK the p-adic completion of A+K [1/π]. Let B = A [1/p],

BK = AK [1/p] and let B denote the completion of the maximal unramified extension of BK in B. SetA = B∩ A, A+ = W (E+), A+ = A+ ∩A and B+ = A+ [1/p] . All these rings are endowed with naturalactions of the Galois group GK and Frobenius ϕ .

Set AK = AHK and BK = AK [1/p] . We remark that Γ and ϕ act on BK by

τ(π) = (1+π)χ(τ)−1, τ ∈ Γ ,

ϕ(π) = (1+π)p−1.

For any r > 0 define

B†,r =

x ∈ B | lim

k→+∞

(ve(xk) +

prp−1

k)

= +∞

.

Set B†,r = B∩ B†,r, B†,rK = BK ∩B†,r, B† =

⋃r>0

B†,r A† = A∩B† and B†K =

⋃r>0

B†,rK .

It can be shown that for any r > p−1

B†,rK =

f (π) = ∑

k∈Zakπ

k | ak ∈ K and f is holomorphic and bounded on B(r,1)

.

Define

B†,rrig,K =

f (π) = ∑

k∈Zakπ

k | ak ∈ K and f is holomorphic on B(r,1)

.

Set R(K) =⋃

r>p−1B†,r

rig,K and R+(K) = R(K)∩K[[π]]. It is not difficult to check that these rings are stable

under Γ and ϕ. To simplify notations we will write R= R(Qp) and R+ = R+(Qp). As usual, we set

62 Denis Benois

t = log(1+π) =∞

∑n=1

(−1)n+1 πn

n∈ R.

Note that ϕ(t) = pt and τ(t) = χ(τ)t, τ ∈ Γ .

1.1.2 (ϕ,Γ )-modules (see [27], [14], [20])

Let A be either B†K or R(K). A (ϕ,Γ )-module over A is a finitely generated free A-module D equipped

with semilinear actions of ϕ and Γ commuting with each other and such that the induced linear map ϕ :A⊗ϕ D→ D is an isomorphism. Such a module is said to be étale if it admits a A†

K-lattice N stable underϕ and Γ and such that ϕ : A†

K ⊗ϕ N → N is an isomorphism. The functor D 7→ R(K)⊗B†K

D induces an

equivalence between the category of étale (ϕ,Γ )-modules over B†K and the category of (ϕ,Γ )-modules

over R(K) which are of slope 0 in the sense of Kedlaya’s theory ([45] and [20], Corollary 1.5). ThenFontaine’s classification of p-adic representations [27] together with the main result of [14] lead to thefollowing statement.

Proposition 1. i) The functorD† : V 7→ D†(V ) = (B†⊗Qp V )HK

establishes an equivalence between the category of p-adic representations of GK and the category of étale(ϕ,Γ )-modules over B†

K .

ii) The functor D†rig(V ) = R(K)⊗B†

KD†(V ) gives an equivalence between the category of p-adic repre-

sentations of GK and the category of (ϕ,Γ )-modules over R(K) of slope 0.

Proof. See [20], Proposition 1.7.

1.1.3 Cohomology of (ϕ,Γ )-modules (see [40], [41], [18])

Fix a generator γ of Γ . If D is a (ϕ,Γ )-module over A, we denote by Cϕ,γ(D) the complex

Cϕ,γ(D) : 0−→D f−→ D⊕D g−→ D−→ 0

where f (x) = ((ϕ − 1)x,(γ − 1)x) and g(y,z) = (γ − 1)y− (ϕ − 1)z. Set H i(D) = H i(Cϕ,γ(D)). A shortexact sequence of (ϕ,Γ )-modules

0−→ D′ −→ D−→ D′′ −→ 0

gives rise to an exact cohomology sequence:

0→ H0(D′)→ H0(D)→ H0(D′′)→ H1(D′)→ ··· → H2(D′′)→ 0.

Proposition 2. Let V be a p-adic representation of GK . Then the complexes RΓ (K,V ), Cϕ,γ(D†(V )) andCϕ,γ(D†

rig(V )) are isomorphic in the derived category of Qp-vector spaces D(Qp).

Proof. This is a derived version of Herr’s computation of Galois cohomology [40]. The proof is given inthe Appendix ( see Proposition A.2 and Corollary A.3).

1.1.4 Iwasawa cohomology

Recall that Λ = Zp[[Γ1]] denotes the Iwasawa algebra of Γ1. Set ∆ = Gal(K1/K) and Λ(Γ ) = Zp[∆ ]⊗Zp Λ .Let ι : Λ(Γ )→Λ(Γ ) denote the involution defined by ι(g) = g−1, g ∈ Γ . If T is a Zp-adic representationof GK , then the induced module IndK∞/K(T ) is isomorphic to (Λ(Γ )⊗Zp T )ι and we set

RΓIw(K,T ) = RΓ (K, IndK∞/K(T )).


Write H iIw(K,T ) for the Iwasawa cohomology

H iIw(K,T ) = lim←−

corKn/Kn−1

H i(Kn,T ).

Recall that there are canonical and functorial isomorphisms

RiΓIw(K,T ) ' H i

Iw(K,T ), i> 0,

RΓIw(K,T )⊗LΛ(Γ )Zp[Gn]' RΓ (Kn,T )

(see [23], Proposition 8.4.22). The interpretation of the Iwasawa cohomology in terms of (ϕ,Γ )-moduleswas found by Fontaine (unpublished but see [15]). We give here the derived version of this result. Letψ : B→ B be the operator defined by the formula ψ(x) = 1

p ϕ−1(TrB/ϕ(B)(x)

). We see immediately that

ψ ϕ = id. Moreover ψ commutes with the action of GK and ψ(A†) = A†. Consider the complexes

CIw,ψ(T ) : D(T )ψ−1−−→ D(T ),

C†Iw,ψ(T ) : D†(T )

ψ−1−−→ D†(T ).

Proposition 3. The complexes RΓIw(K,T ), CIw,ψ(T ) and C†Iw,ψ(T ) are naturally isomorphic in the derived

category D(Λ(Γ )) of Λ(Γ )-modules.

Proof. See Proposition A.5 and Corollary A.6.

1.1.5 (ϕ,Γ )-modules of rank 1

Recall the computation of the cohomology of (ϕ,Γ )-modules of rank 1 following Colmez [20]. As inop. cit., we consider the case K = Qp and put R = B†

rig,Qpand R+ = B+

rig,Qp. The differential operator

∂ = (1+π)d

dπacts on R and R+. If δ : Q∗p→Q∗p is a continuous character, we write R(δ ) for the (ϕ,Γ )-

module Reδ defined by ϕ(eδ ) = δ (p)eδ and γ(eδ ) = δ (χ(τ))eδ . Let x denote the character induced by thenatural inclusion of Qp in L and |x| the character defined by |x|= p−vp(x).

Proposition 4. Let δ : Q∗p→Q∗p be a continuous character. Then:i)

H0(R(δ )) =

Qptm if δ = x−m,m ∈ N0 otherwise.

ii)

dimQp(H1(R(δ ))) =

2 if either δ (x) = x−m,m> 0 or δ (x) = |x|xm,m> 1,1 otherwise.

iii) Assume that δ (x) = x−m, m> 0. The classes cl(tm,0)eδ and cl(0, tm)eδ form a basis of H1(R(x−m)).iv) Assume that δ (x) = |x|xm, m> 1. Then H1(R(|x|xm)), m> 1 is generated by cl(αm) and cl(βm) where

αm =(−1)m−1

(m−1)!∂

m−1(

1π+

12,a)

eδ , (1−ϕ)a = (1−χ(γ)γ)

(1π+

12

),

βm =(−1)m−1

(m−1)!∂

m−1(

b,1π

)eδ , (1−ϕ)

(1π

)= (1−χ(γ)γ)b

Proof. See [20], Sect. 2.3-2.5.

64 Denis Benois

1.2 Crystalline representations

1.2.1 The rings Bcris and BdR (see [26], [29])

Let θ0 : A+→ OC be the map given by the formula

θ0

(∞

∑n=0

[un]pn

)=

∞

∑n=0

u(0)n pn.

It can be shown that θ0 is a surjective ring homomorphism and that ker(θ0) is the principal ideal generated

by ω =p−1∑

i=0[ε]i/p. By linearity, θ0 can be extended to a map θ : B+→C. The ring B+

dR is defined to be the

completion of B+ for the ker(θ)-adic topology:

B+dR = lim←−

nB+/ker(θ)n.

This is a complete discrete valuation ring with residue field C equipped with a natural action of GK . More-

over, there exists a canonical embedding K⊂B+dR. The series t =

∞

∑n=0

(−1)n−1πn/n converges in the topology

of B+dR and it is easy to see that t generates the maximal ideal of B+

dR. The Galois group acts on t by theformula g(t) = χ(g)t. Let BdR = B+

dR[t−1] be the field of fractions of B+

dR. This is a complete discrete valu-ation field equipped with a GK-action and an exhaustive separated decreasing filtration FiliBdR = t iB+

dR. AsGK-modules, FiliBdR/Fili+1BdR 'C(i) and BGK

dR = K.Consider the PD-envelope of A+ with a respect to the map θ0

APD = A+

[ω2

2!,

ω3

3!, . . . ,

ωn

n!, . . .

]and denote by A+

cris its p-adic completion. Let B+cris = A+

cris ⊗Zp Qp and Bcris = B+cris[t

−1]. Then Bcris isa subring of BdR endowed with the induced filtration and Galois action. Moreover, it is equipped with acontinuous Frobenius ϕ , extending the map ϕ : A+→ A+. One has ϕ(t) = pt.

1.2.2 Crystalline representations (see [30], [7], [8])

Let L be a finite extension of Qp. Denote by K its maximal unramified subextension. A filtered Dieudonnémodule over L is a finite dimensional K-vector space M equipped with the following structures:• a σ -semilinear bijective map ϕ : M→M;• an exhaustive decreasing filtration (FiliML)i∈Z on the L-vector space ML = L⊗K M.A K-linear map f : M→M′ is said to be a morphism of filtered modules if• f (ϕ(d)) = ϕ( f (d)) for all d ∈M;• f (FiliML)⊂ FiliM′L for all i ∈ Z.The category MFϕ

L of filtered Dieudonné modules is additive, has kernels and cokernels but is not abelian.Denote by 1 the vector space K0 with the natural action of σ and the filtration given by

Fili1 =

K, if i6 0,0, ifi > 0.

Then 1 is a unit object of MFϕ

L i.e. M⊗1' 1⊗M 'M for any M.If M is a one dimensional Dieudonné module and d is a basis vector of M, then ϕ(d) = αd for some

α ∈ K. Set tN(M) = vp(α) and denote by tH(M) the unique filtration jump of M. If M is of an arbitraryfinite dimension d, set tN(M) = tN(∧dM) and tH(M) = tH(∧dM). A Dieudonné module M is said to beweakly admissible if tH(M) = tN(M) and if tH(M′)6 tN(M′) for any ϕ-submodule M′ ⊂M equipped withthe induced filtration. Weakly admissible modules form a subcategory of MFϕ

L which we denote by MFϕ, fL .


If V is a p-adic representation of GL, define DdR(V ) = (BdR⊗V )GL . Then DdR(V ) is a L-vector spaceequipped with the decreasing filtration FiliDdR(V ) = (FiliBdR⊗V )GL . One has dimL DdR(V ) 6 dimQp(V )

and V is said to be de Rham if dimL DdR(V ) = dimQp(V ). Analogously one defines Dcris(V ) = (Bcris⊗V )GL .Then Dcris(V ) is a filtered Dieudonné module over L of dimension dimK Dcris(V ) 6 dimQp(V ) and Vis said to be crystalline if the equality holds here. In particular, for crystalline representations one hasDdR(V ) = Dcris(V )⊗K L. By the theorem of Colmez–Fontaine [21], the functor Dcris establishes an equiva-lence between the category of crystalline representations of GL and MFϕ, f

L . Its quasi-inverse Vcris is givenby Vcris(D) = Fil0(D⊗K Bcris)

ϕ=1.An important result of Berger ([7], Theorem 0.2) says that Dcris(V ) can be recovered from the (ϕ,Γ )-

module D†rig(V ). The situation is particularly simple if L/Qp is unramified. In this case set D+(V ) = (V ⊗Qp

B+)HK and D+rig(V ) = R+(K)⊗B+

KD+(V ). Then

Dcris(V ) =

(D+

rig(V )

[1t

])Γ

(see [8], Proposition 3.4).

2 The exponential map

2.1 The Bloch–Kato exponential map ([4], [22], [31])

2.1.1 Cohomology of Dieudonné modules

Let L be a finite extension of Qp and K its maximal unramified subextension. Recall that we denote by MFϕ

Lthe category of filtered Dieudonné modules over L. If M is an object of MFϕ

L , define

H i(L,M) = ExtiMFϕ

L(1,M), i = 0,1.

We remark that H∗(L,M) can be computed explicitly as the cohomology of the complex

C•(M) : Mf−→ (ML/Fil0ML)⊕M

where the modules are placed in degrees 0 and 1 and f (d) = (d (mod Fil0ML),(1−ϕ)(d)) (see [22],[31]).Note that if M is weakly admissible then each extension 0→M→M′→ 1→ 0 is weakly admissible tooand we can write H i(L,M) = Exti

MFϕ, fL

(1,M).

2.1.2 The exponential map

Let Repcris(GL) denote the category of crystalline representations of GL. For any object V of Repcris(GL)define

H if (L,V ) = ExtiRepcris(GL)

(Qp(0),V ).

An easy computation shows that

H if (L,V ) =

H0(L,V ) if i = 0,ker(H1(L,V )→ H1(L,V ⊗Bcris)) if i = 1,0 if i> 2.

Let tV (L) = DdR(V )/Fil0DdR(V ) denote the tangent space of V . The rings BdR and Bcris are related to eachother via the fundamental exact sequence

66 Denis Benois

0−→Qp −→ Bcrisf−→ BdR/Fil0BdR⊕Bcris −→ 0

where f (x) = (x (mod Fil0BdR),(1−ϕ)x) (see [4], §4). Tensoring this sequence with V and taking coho-mology one obtains an exact sequence

0−→ H0(L,V )−→ Dcris(V )−→ tV (L)⊕Dcris(V )−→ H1f (L,V )−→ 0.

The last map of this sequence gives rise to the Bloch–Kato exponential map

expV,L : tV (L)⊕Dcris(V )−→ H1(L,V ).

Following [28] set

RΓf (L,V ) = C•(Dcris(V )) =[Dcris(V )

f−→ tV (L)⊕Dcris(V )].

From the classification of crystalline representations in terms of Dieudonné modules it follows that thefunctor Vcris induces natural isomorphisms

riV,p : Ri

Γf (L,V )−→ H if (L,V ), i = 0,1.

The composite homomorphism

tV (L)⊕Dcris(V )−→ R1Γf (L,V )

r1V,p−→ H1(L,V )

coincides with the Bloch–Kato exponential map expV,L ([22], Proposition 1.21).

2.1.3 The map RΓf (L,V )→ RΓ (L,V )

Let g : B•→C• be a morphism of complexes. We denote by Tot•(g) the complex Totn(g) =Cn−1⊕Bn withdifferentials dn : Totn(g)→ Totn+1(g) defined by the formula dn(c,b) = ((−1)ngn(b)+ dn−1(c),dn(b)).

It is well known that if 0 −→ A•f−→ B•

g−→C•−→0 is an exact sequence of complexes, then f induces aquasi isomorphism A• ∼→ Tot•(g). In particular, tensoring the fundamental exact sequence with V, we obtainan exact sequence of complexes

0→ RΓ (L,V )→C•c (GL,V ⊗Bcris)f→

C•c (GL,(V ⊗ (BdR/Fil0BdR))⊕ (V ⊗Bcris))→ 0

which gives a quasi isomorphism RΓ (L,V )∼→ Tot•( f ). Since RΓf (L,V ) coincides tautologically with the

complex

C0c (GL,V ⊗Bcris)

f−→C0c (GL,(V ⊗ (BdR/Fil0BdR))⊕ (V ⊗Bcris))

we obtain a diagramRΓ (L,V )

∼ // Tot•( f )

RΓf (L,V )

OOff

which defines a morphism RΓf (L,V )→ RΓ (L,V ) in D(Qp) (see [12], Sect. 1.2.1). We remark that theinduced homomorphisms RiΓf (L,V )→H i(L,V ) (i= 0,1) coincide with the composition of ri

V,p with naturalembeddings H i

f (L,V )→ H i(L,V ).


2.1.4 Exponential map for (ϕ,Γ )-modules

In this subsection we define an analogue of the exponential map for crystalline (ϕ,Γ )-modules. See [51]for a more general setting. Let K/Qp be an unramified extension. If D is a (ϕ,Γ )-module over R(K) define

Dcris(D) = (D [1/t])Γ .

It can be shown that Dcris(D) is a finite dimensional K-vector space equipped with a natural decreasingfiltration FiliDcris(D) and a semilinear action of ϕ . One says that D is crystalline if

dimK(Dcris(D)) = rank(D).

From [10], Théorème A it follows that the functor D 7→ Dcris(D) is an equivalence between the categoryof crystalline (ϕ,Γ )-modules and MFϕ

K . Remark that if V is a p-adic representation of GK then Dcris(V ) =

Dcris(D†rig(V )) and V is crystalline if and only if D†

rig(V ) is.Let D be a (ϕ,Γ )-module. To any cocycle α = (a,b) ∈ Z1(Cϕ,γ(D)) one can associate the extension

0−→ D−→ Dα −→ R(K)−→ 0

defined byDα = D⊕R(K)e, (ϕ−1)e = a, (γ−1)e = b.

As usual, this gives rise to an isomorphism H1(D) ' Ext1R(R(K),D). We say that cl(α) is crystalline ifdimK (Dcris(Dα)) = dimK (Dcris(D))+1 and define

H1f (D) = cl(α) ∈ H1(D) | cl(α) is crystalline

(see [3], Sect. 1.4.1). If D is crystalline (or more generally potentially semistable) one has a natural isomor-phism

H1(K,Dcris(D))−→ H1f (D).

Set tD =Dcris(D)/Fil0Dcris(D) and denote by

expD : tD⊕Dcris(D)→ H1(D)

the composition of this isomorphism with the projection

tD⊕Dcris(D)→ H1(K,Dcris(D))

and the embedding H1f (D) → H1(D).

Assume that K =Qp. To simplify notation we will write Dm for R(|x|xm) and em for its canonical basis.Then Dcris(Dm) is the one dimensional Qp-vector space generated by t−mem. As in [3], we normalise thebasis (cl(αm),cl(βm)) of H1(Dm) putting α∗m = (1−1/p) cl(αm) and β ∗m = (1−1/p) log(χ(γ))cl(βm).

Proposition 5. i) H1f (Dm) is the one-dimensional Qp-vector space generated by α∗m.

ii) The exponential mapexpDm

: tDm −→ H1(Dm)

sends t−mwm to −α∗m.

Proof. This is a reformulation of [3], Proposition 1.5.8 ii).

2.2 The large exponential map

2.2.1 Notation

In this section p is an odd prime number, K is a finite unramified extension of Qp and σ the absolute Frobe-nius acting on K. Recall that Kn = K(ζpn) and K∞ =

⋃∞n=1 Kn. We set Γ = Gal(K∞/K), Γn = Gal(K∞/Kn)

68 Denis Benois

and ∆ = Gal(K1/K). Let Λ = Zp[[Γ1]] and Λ(Γ ) = Zp[∆ ]⊗Zp Λ . We will consider the following operatorsacting on the ring K[[X ]] of formal power series with coefficients in K:• The ring homomorphism σ : K[[X ]]−→ K[[X ]] defined by

σ

(∞

∑i=0

aiX i

)=

∞

∑i=0

σ(ai)X i.

• The ring homomorphism ϕ : K[[X ]]−→ K[[X ]] defined by

ϕ

(∞

∑i=0

aiX i

)=

∞

∑i=0

σ(ai)ϕ(X)i, ϕ(X) = (1+X)p−1.

• The differential operator ∂ = (1+X)d

dX. One has ∂ ϕ = pϕ ∂ .

• The operator ψ : K[[X ]]−→ K[[X ]] defined by

ψ( f (X)) =1p

ϕ−1

(∑

ζ p=1f ((1+X)ζ −1)

).

It is easy to see that ψ is a left inverse to ϕ, i.e. that ψ ϕ = id.

• An action of Γ given by γ

(∞

∑i=0

aiX i

)=

∞

∑i=0

aiγ(X)i, γ(X) = (1+X)χ(γ)−1.

We remark that these formulas are compatible with the definitions fromSect. 1.1.1 and 1.1.4. Fix a generator γ1 ∈ Γ1 and define

H = f (γ1−1) | f ∈Qp[[X ]] is holomorphic on B(0,1),H(Γ ) = Zp[∆ ]⊗Zp H.

2.2.2 The map ΞΞΞεV,n

It is well known that Zp[[X ]]ψ=0 is a free Λ -module generated by (1+X) and the operator ∂ is bijectiveon Zp[[X ]]ψ=0. If V is a crystalline representation of GK put D(V ) = Dcris(V )⊗Zp Zp[[X ]]ψ=0. Let ΞΞΞ

εV,n :

D(V )Γn [−1]−→ RΓf (Kn,V ) be the map defined by

ΞΞΞεV,n(α) =

p−n(∑n

k=1(σ ⊗ϕ)−kα(ζpk −1),−α(0)) if n> 1,TrK1/K

(ΞΞΞ

εV,1(α)

)if n = 0.

An easy computation shows that ΞΞΞεV,0 : Dcris(V )[−1]→ RΓf (K,V ) is given by the formula

ΞΞΞεV,0(a) =

1p(−ϕ

−1(a),−(p−1)a).

In particular, it is homotopic to the map a 7→ −(0,(1− p−1ϕ−1)a). Let

ΞεV,n : D(V )→ R1

Γ (Kn,V ) =tV (Kn)⊕Dcris(V )

Dcris(V )/V GK

denote the homomorphism induced by ΞΞΞεV,n. Then

ΞεV,0(a) =−(0,(1− p−1

ϕ−1)a) (mod Dcris(V )/V GK ).

If Dcris(V )ϕ=1 = 0 the operator 1−ϕ is invertible on Dcris(V ) and we can write


ΞεV,0(a) =

(1− p−1ϕ−1

1−ϕa,0)

(mod Dcris(V )/V GK ). (3)

For any i ∈ Z let ∆i : D(V )→ Dcris(V )

(1− piϕ)Dcris(V )⊗Qp(i) be the map given by

∆i(α(X)) = ∂iα(0)⊗ ε

⊗i (mod (1− piϕ)Dcris(V )).

Set ∆ = ⊕i∈Z∆i. If α ∈ D(V )∆=0, then by [25], Proposition 2.2.1 there exists F ∈ Dcris(V )⊗Qp Qp[[X ]]which converges on the open unit disk and such that(1−ϕ)F = α. A short computation shows that

ΞεV,n(α) = p−n((σ ⊗ϕ)−n(F)(ζpn −1),0) (mod Dcris(V )/V GK ), if n> 1

(see [BB12], Lemme 4.9).

2.2.3 Construction of the large exponential map

As Zp[[X ]] [1/p] is a principal ideal domain and H is Zp[[X ]] [1/p]-torsion free, H is flat. Thus

C†Iw,ψ(V )⊗L

ΛQpH(Γ ) = C†

Iw,ψ(V )⊗ΛQpH(Γ ) =

=[H(Γ )⊗ΛQp

D†(V )ψ−1−−→H(Γ )⊗ΛQp

D†(V )].

By Proposition 3 one has an isomorphism in D(H(Γ ))

RΓIw(K,V )⊗LΛQp

H(Γ )'C†Iw,ψ(V )⊗ΛQp

H(Γ ).

The action of H(Γ ) on D†(V )ψ=1 induces an injection H(Γ )⊗ΛQpD†(V )ψ=1 → D†

rig(V )ψ=1. Composingthis map with the canonical isomorphism H1

Iw(K,V )'D†(V )ψ=1 we obtain a map H(Γ )⊗ΛQpH1

Iw(K,V ) →

D†rig(V )ψ=1. For any k ∈Z set ∇k = t∂ −k = t

ddt−k. An easy induction shows that ∇k−1 ∇k−2 · · ·∇0 =

tk∂ k.Fix h> 1 such that Fil−hDcris(V ) = Dcris(V ) and V (−h)GK = 0. For any α ∈D(V )∆=0 define

ΩεV,h(α) = (−1)h−1 log χ(γ1)

p∇h−1 ∇h−2 · · ·∇0(F(π)),

where F ∈ H(V ) is such that (1−ϕ)F = α. It is easy to see that Ω εV,h(α) ∈ D+

rig(V )ψ=1. In [9] Bergershows that Ω ε

V,h(α) ∈H(Γ )⊗ΛQpD†(V )ψ=1 and therefore gives rise to a map

ExpεV,h : D(V )∆=0[−1]→ RΓIw(K,V )⊗L

ΛQpH(Γ )

LetExpε

V,h : D(V )∆=0→H(Γ )⊗ΛQpH1

Iw(K,V )

denote the map induced by ExpεV,h in degree 1. The following theorem is a reformulation of the construction

of the large exponential map given by Berger in [9].

Theorem 3. LetExpε

V,h,n : D(V )∆=0Γn

[−1]−→ RΓIw(K,V )⊗LΛQp

Qp[Gn].

denote the map induced by ExpεV,h. Then for any n> 0 the following diagram in D(Qp[Gn]) is commutative:

70 Denis Benois

D(V )∆=0Γn

[−1]Expε

V,h,n //

ΞΞΞεV,n

RΓIw(K,V )⊗LΛQp

Qp[Gn]

'

RΓf (Kn,V )(h−1)! // RΓ (Kn,V ) .

In particular, ExpεV,h coincides with the large exponential map of Perrin-Riou.

Proof. Passing to cohomology in the previous diagram one obtains the diagram

D(V )∆=0Expε

V,h //

ΞεV,n

H(Γ )⊗ΛQpH1

Iw(K,V )

prV,n

DdR/Kn(V )⊕Dcris(V )(h−1)! expV,Kn // H1(Kn,V )

which is exactly the definition of the large exponential map. Its commutativity is proved in [9], TheoremII.13. Now, the theorem is an immediate consequence of the following remark. Let D be a free A-moduleand let f1, f2 : D[−1]→ K• be two maps from D[−1] to a complex of A-modules such that the inducedmaps H1( f1) and H1( f2) : D→ H1(K•) coincide. Then f1 and f2 are homotopic.

remark The large exponential map was first constructed in [25]. See [17] and [2] for alternative constructionsand [26], [51] and [60] for generalisations.

3 The L -invariant

3.1 Definition of the L -invariant

3.1.1 Preliminaries

Let S be a finite set of primes of Q containing p and GS the Galois group of the maximal algebraic extensionof Q unramified outside S∪∞. For each place v we denote by Gv the decomposition group at v, by Qur

vthe maximal unramified extension of Qv and by Iv and fv the inertia subgroup and Frobenius automorphismrespectively. Let V be a pseudo-geometric p-adic representation of GS. This means that the restriction of Von the decomposition group at p is a de Rham representation. Following Greenberg, for any v /∈ p,∞ set

RΓf (Qv,V ) =[V Iv 1− fv−−−→V Iv

],

where the terms are placed in degrees 0 and 1 (see [28], [12]). We remark that there is a naturalquasi-isomorphism RΓf (Qv,V )'C•c (Gv/Iv,V Iv). Note that R0Γf (Qv,V ) = H0(Qv,V ) and R1Γf (Qv,V ) =H1

f (Qv,V ) whereH1

f (Qv,V ) = ker(H1(Qv,V )→ H1(Qurv ,V )).

For v= p the complex RΓf (Qv,V ) was defined in Sect. 2.1.2. To simplify notation write H iS(V ) =H i(GS,V )

for the continuous Galois cohomology of GS with coefficients in V . The Bloch–Kato Selmer group of V isdefined as

H1f (V ) = ker

(H1

S (V )→⊕v∈S

H1(Qv,V )

H1f (Qv,V )

).

We also set

H1f ,p(V ) = ker

H1S (V )→

⊕v∈S−p

H1(Qv,V )

H1f (Qv,V )

.


From the Poitou–Tate exact sequence one obtains the following exact sequence relating these groups (seefor example [55], Lemme 3.3.6)

0→ H1f (V )→ H1

f ,p(V )→H1(Qp,V )

H1f (Qp,V )

→ H1f (V∗(1)) .

We also have the following formula relating dimensions of Selmer groups (see [31], II, 2.2.2)

dimQp H1f (V )−dimQp H1

f (V∗(1))−dimQp H0

S (V )+dimQp H0S (V

∗(1)) =

dimQp tV (Qp)−dimQp H0(R,V ).

Set d±(V ) = dimQp(Vc=±1), where c denotes complex conjugation.

3.1.2 Basic assumptions

Assume that V satisfies the following conditionsC1) H1

f (V∗(1)) = 0.

C2) H0S (V ) = H0

S (V∗(1)) = 0.

C3) V is crystalline at p and ϕ : Dcris(V )→ Dcris(V ) is semisimple at 1 and p−1.C4) Dcris(V )ϕ=1 = 0.C5) The localisation map

locp : : H1f (V )→ H1

f (Qp,V )

is injective.

These conditions appear naturally in the following situation. Let X be a proper smooth variety over Q.Let H i

p(X) denote the p-adic étale cohomology of X . Consider the Galois representations V = H ip(X)(m).

By Poincaré duality together with the hard Lefschetz theorem we have

H ip(X)∗ ' H i

p(X)(i)

and thus V ∗(1)'V (i+1−2m). The Beilinson conjecture (in the formulation of Bloch and Kato) predictsthat

H1f (V∗(1)) = 0 if w6−2.

This corresponds to the hope that there are no nontrivial extensions of Q(0) by motives of weight > 0.If X has a good reduction at p, then V is crystalline [25] and the semisimplicity of ϕ is a well known(and difficult) conjecture. By a result of Katz and Messing [44] Dcris(V )ϕ=1 6= 0 can occur only if i = 2m.Therefore up to eventually replace V by V ∗(1) the conditions C1, C3-4) conjecturally hold except the weight−1 case i = 2m−1.

The condition Dcris(V )ϕ=1 = 0 implies that the exponential map tV (Qp)→H1f (Qp,V ) is an isomorphism

and we denote by logV its inverse. The composition of the localisation map locp with the Bloch–Katologarithm

rV : H1f (V )→ tV (Qp)

coincides conjecturally with the p-adic (syntomic) regulator. We remark that if H0(Qv,V ) = 0 for all v 6= p(and therefore H1

f (Qv,V ) = 0 for all v 6= p) then locp is injective for all m 6= i/2, i/2+ 1 by a result ofJannsen ([42], Lemma 4 and Theorem 3).

If H0S (V ) 6= 0, then V contains a trivial subextension V0 = Qp(0)k. For Qp(0) our theory describes the

behavior of the Kubota–Leopoldt p-adic L-function and is well known. Therefore we can assume thatH0

S (V ) = 0. Applying the same argument to V ∗(1) we can also assume that H0S (V

∗(1)) = 0.

From our assumptions we obtain an exact sequence

0→ H1f (V )→ H1

f ,p(V )→H1(Qp,V )

H1f (Qp,V )

→ 0. (4)

72 Denis Benois

Moreover

dimQp H1f (V ) = dimQp tV (Qp)−d+(V ), (5)

dimQp H1f ,p(V ) = d−(V )+dimQp H0(Qp,V ∗(1)). (6)

3.1.3 Regular submodules

In the remainder of this section we assume that V satisfies C1-5).

Definition (PERRIN-RIOU). 1) A ϕ-submodule D of Dcris(V ) is regular if D∩Fil0Dcris(V ) = 0 and the map

rV,D : H1f (V )→ Dcris(V )/(Fil0Dcris(V )+D)

induced by rV is an isomorphism.2) Dually, a ϕ-submodule D of Dcris(V ∗(1)) is regular if

D+Fil0Dcris(V ∗(1)) = Dcris(V ∗(1))

and the mapD∩Fil0Dcris(V ∗(1))→ H1(V )∗

induced by the dual map r∗V : Fil0Dcris(V ∗(1))→ H1(V )∗ is an isomorphism.

It is easy to see that if D is a regular submodule of Dcris(V ), then

D⊥ = Hom(Dcris(V )/D,Dcris(Qp(1))

is a regular submodule of Dcris(V ∗(1)). From (6) we also obtain that

dimD = d+(V ), dimD⊥ = d−(V ) = d+(V ∗(1)).

Let D⊂Dcris(V ) be a regular submodule. As in [3] we use the semisimplicity of ϕ to decompose D intothe direct sum

D = D−1⊕Dϕ=p−1.

which gives a four step filtration0 ⊂ D−1 ⊂ D⊂ Dcris(V ).

Let D0 = D and D−1 denote the (ϕ,Γ )-submodules associated to D and D−1 by Berger’s theory, thus

D = Dcris(D), D−1 = Dcris(D−1).

Set W = gr0D†rig(V ). Thus we have two tautological exact sequences

0→ D→ D†rig(V )→ D′→ 0,

0→ D−1→ D→W → 0.

Note the following properties of cohomology of these modules:a) The natural maps H1(D−1)→H1(D) and H1(D)→H1(D†

rig(V )) = H1(Qp,V ) are injective. This fol-lows from the observation that Dcris(D′)ϕ=1 = 0 by C4). Since H0(D′) = Fil0Dcris(D′)ϕ=1 ([3], Proposition1.4.4) we have H0(D′) = 0. The same argument works for W .

b) H1f (D−1) = H1(D−1). In particular the exponential map expD−1

: D−1 → H1(D−1) is an isomor-phism. This follows from the computation of dimensions of H1(D−1) and H1

f (D−1). Namely, since

Dϕ=1−1 = Dϕ=p−1

−1 = 0 the Euler–Poincaré characteristic formula [18] together with Poincaré duality give


dimQp H1(D−1) =

= rank(D−1)−dimQp H0(D−1)−dimQp H0(D∗−1(χ)) = dimQp(D−1).

On the other hand sinceFil0D−1 = D−1∩Fil0Dcris(V ) = 0

one has dimQp H1f (D−1) = dimQp(D−1) by [3], Corollary 1.4.5.

c) The exponential map expD : D→ H1f (D) is an isomorphism. This follows from Fil0D = 0 and

Dϕ=1 = 0.The regularity of D is equivalent to the decomposition

H1f (Qp,V ) = H1

f (V )⊕H1f (D). (7)

Since locp is injective by C5), the localisation map H1f ,p(V )→ H1(Qp,V ) is also injective. Let

κD : H1f ,p(V )→

H1(Qp,V )

H1f (D)

denote the composition of this map with the canonical projection.

Lemma 1. i) One hasH1

f (Qp,V )∩H1(D) = H1f (D).

ii) κD is an isomorphism.

Proof. i) Since H0(D′) = 0 we have a commutative diagram with exact rows and injective colomns

0 // H1f (D) //

H1f (Qp,V ) //

H1f (D

′)

0 // H1(D) // H1(Qp,V ) // H1(D′).

This gives i).ii) Since H1

f (D) ⊂ H1f (Qp,V ) one has ker(κD) ⊂ H1

f (Qp,V ). One the other hand (7) shows that κD isinjective on H1

f (V ). Thus ker(κD) = 0. On the other hand, because dimQp H1f (D) = dimQp(D) we have

dimQp

(H1(Qp,V )

H1f (D)

)= d−(V )+dimQp H0(Qp,V ∗(1)).

Comparing this with (5-6) we obtain that κD is an isomorphism.

3.1.4 The main construction

Set e = dimQp(Dϕ=p−1

). The (ϕ,Γ )-module W satisfies

Fil0Dcris(W ) = 0, Dcris(W )ϕ=p−1= Dcris(W ).

(Recall that Dcris(W ) = Dϕ=p−1.) The cohomology of such modules was studied in detail in [3], Proposition

1.5.9 and Sect. 1.5.10. Namely, H0(W )= 0, dimQp H1(W )= 2e and dimQp(W )= e. There exists a canonicaldecomposition

H1(W ) = H1f (W )⊕H1

c (W )

of H1(W ) into the direct sum of H1f (W ) and some canonical space H1

c (W ). Moreover there exist canonicalisomorphisms

iD, f : Dcris(W )' H1f (W ), iD,c : Dcris(W )' H1

c (W ).

74 Denis Benois

These isomorphisms can be described explicitly. By Proposition 1.5.9 of [3]

W 'e⊕

i=1Dmi ,

where Dmi = R(|x|xmi), mi > 1. By Proposition 5 H1f (Dm) is generated by α∗m and H1

c (Dm) is the subspacegenerated by β ∗m (see also Proposition 1.1.9). Then

iD, f (x) = xα∗m, iD,c(x) = xβ

∗m.

Since H0(W ) = 0 and H2(D−1) = 0 we have exact sequences

0→ H1(D−1)→ H1(D)→ H1(W )→ 0,0→ H1

f (D−1)→ H1f (D)→ H1

f (W )→ 0.

Since H1f (D−1) = H1(D) we obtain that

H1(D)

H1f (D)

' H1(W )

H1f (W )

.

Let H1(D,V ) denote the inverse image of H1(D)/H1f (D) by κD. Then κD induces an isomorphism

H1(D,V )' H1(D)

H1f (D)

.

By Lemma 1 the localisation map H1(D,V )→ H1(W ) is well defined and injective. Hence, we have adiagram

Dcris(W )iD, f∼ // H1

f (W )

H1(D,V )

ρD, f

OO

//

ρD,c

H1(W )

pD, f

OO

pD,c

Dcris(W )

iD,c∼ // H1c (W ),

where ρD, f and ρD,c are defined as the unique maps making this diagram commute. From Lemma 1 ii) itfollows that ρD,c is an isomorphism. The following definition generalises (in the crystalline case) the mainconstruction of [3] where we assumed in addition that H1

f (V ) = 0.

Definition The determinantL (V,D) = det

(ρD, f ρ

−1D,c | Dcris(W )

)will be called the L -invariant associated to V and D.

3.2 L -invariant and the large exponential map

3.2.1 Differentiation of the large exponential map

In this section we interpret L (V,D) in terms of the derivative of the large exponential map. This inter-pretation is crucial for the proof of the main theorem of this paper. Recall that H1(Qp,H(Γ )⊗Qp V ) =

H(Γ )⊗Λ(Γ ) H1Iw(Qp,V ) injects into D†

rig(V ). Set


F0H1(Qp,H(Γ )⊗Qp V ) = D∩H1(Qp,H(Γ )⊗Qp V ),

F−1H1(Qp,H(Γ )⊗Qp V ) = D−1∩H1(Qp,H(Γ )⊗Qp V ).

As in Sect. 2.2 we fix a generator γ ∈ Γ . The following result is a straightforward generalisation of [4],Prop. 2.2.2. For the convenience of the reader we give here the proof which is the same as in op. cit. moduloobvious modifications.

Proposition 6. Let D be a regular subspace of Dcris(V ). For any a ∈ Dϕ=p−1let α ∈ D(V ) be such that

α(0) = a. Theni) There exists a unique β ∈ F0H1(Qp,H(Γ )⊗V ) such that

(γ−1)β = ExpεV,h(α).

ii) The composite map

δD,h : Dϕ=p−1 → F0H1(Qp,H(Γ )⊗V )→ H1(W )

δD,h(a) = β (mod H1(D−1))

is given explicitly by the following formula:

δD,h(a) = −(h−1)!(

1− 1p

)−1

(log χ(γ))−1 iD,c(α).

Proof. Since Dcris(V )ϕ=1 = 0, the operator 1−ϕ is invertible on Dcris(V ) and we have a diagram

D(V )∆=0Expε

V,h //

Ξ εV,0

H1(Qp,H(Γ )⊗V )

prV

Dcris(V )(h−1)!expV // H1(Qp,V ).

where Ξ εV,0(α)=

1− p−1ϕ−1

1−ϕα(0) (see (3)). If α ∈Dϕ=p−1⊗Zp[[X ]]ψ=0, then Ξ ε

V,0(α)= 0 and prV

(Expε

V,h(α))=

0. On the other hand, as V GK = 0 the map(H(Γ )⊗ΛQp

H1Iw(Qp,V )

)Γ

→H1(Qp,V ) is injective. Thus there

exists a unique β ∈H(Γ )⊗Λ H1Iw(Qp,T ) such that Expε

V,h(α) = (γ−1)β . Now take a ∈ Dϕ=p−1and set

f = a⊗ `

((1+X)χ(γ)−1

X

),

where `(g) =1p

log(

gp

ϕ(g)

). An easy computation shows that

∑ζ p=1

`

(ζ χ(γ)(1+X)χ(γ)−1

ζ (1+X)−1

)= 0.

Thus f ∈ Dϕ=p−1 ⊗Zp[[X ]]ψ=0. Write α in the form α = (1−ϕ)(1− γ)(a⊗ log(X)). Then

ΩV,h(α) = (−1)h−1 log χ(γ1)

pth

∂h((γ−1)(a log(π)) =

log χ(γ1)

p(γ−1)β

where

β = (−1)h−1th∂

h(a log(π)) = (−1)h−1ath∂

h−1(

1+π

π

).

76 Denis Benois

This implies immediately that β ∈ D. On the other hand Dϕ=p−1= Dcris(W ) = (W [1/t])Γ and we will

write a for the image of a in W [1/t] . By [3], Sect. 1.5.8-1.5.10 one has W 'e⊕

i=1Dmi where Dm = R(|x|xm)

and we denote by em the canonical base of Dm. Then without loss of generality we may assume thata = t−miemi for some i. Let β be denote the image of β in W ψ=1 and let h1

0 : W ψ=1 → H1(W ) bethe canonical map furnished by Proposition 3. Recall that h1

0(β ) = cl(c, β ) where (1− γ)c = (1−ϕ) β .

Then β = (−1)h−1th−mi∂ h log(π). By Lemma 1.5.1 of [14] there exists a unique b0 ∈ B†,ψ=0Qp

such that(γ−1)b0 = `(π). This implies that

(1− γ)(th−mi∂hb0emi) = (1−ϕ)(th−mi∂

h log(π)emi) = (−1)h−1(1−ϕ) β .

Thus c = (−1)h−1th−mi∂ hb0emi and res(ctmi−1dt) = (−1)h−1res(th−1∂ hb0dt)emi = 0. Next from the con-gruence β ≡ (h−1)! t−miemi (mod Qp[[π]]emi) it follows that res(β tmi−1dt) = (h−1)!emi . Therefore by[3], Corollary 1.5.6 we have

cl(c, β ) = (h−1)!cl(βm) = (h−1)!p

log χ(γ1)iW,c(a).

On the other hand

α(0) = a⊗ `

((1+X)χ(γ)−1

X

)∣∣∣∣∣X=0

= a(

1− 1p

)log(χ(γ)).

These formulas imply that

δD,h(a) = (h−1)!(

1− 1p

)−1

(log χ(γ))−1 iW,c(a).

and the proposition is proved.

3.2.2 Interpretation of the L -invariant

From the definition of H1(D,V ) and Lemma 1 we immediately obtain that

H1(Qp,V )

H1f ,p(V )+H1(D−1)

' H1(D)

H1(D,V )+H1(D−1)' H1(W )

H1(D,V ).

Thus, the map δD,h constructed in Proposition 6 induces a map

Dϕ=p−1 →H1(Qp,V )

H1f ,p(V )+H1(D−1)

which we will denote again by δD,h. On the other hand, we have isomorphisms

Dϕ=p−1 expV−−→H1

f (Qp,V )

expV,Qp(D−1)

'H1

f (Qp,V )

H1(D−1)'

H1(Qp,V )

H1f ,p(V )+H1(D−1)

.

Proposition 7. Let λD : Dϕ=p−1 → Dϕ=p−1denote the homomorphism making the diagram

Dϕ=p−1

δD,h

&&

λD // Dϕ=p−1

(h−1)!expV

xxH1(Qp,V )

H1f ,p(V )+H1(D−1)


commute. Then

det(

λD |Dϕ=p−1)= (log χ(γ))−e

(1− 1

p

)−e

L (V,D).

Proof. The proposition follows from Proposition 6 and the following elementary fact. Let U =U1⊕U2 bethe decomposition of a vector space U of dimension 2e into the direct sum of two subspaces of dimensione. Let X ⊂U be a subspace of dimension e such that X ∩U1 = 0. Consider the diagrams

Xp1 //

p2

U1 U/X U1i1oo

U2

f

>>

U2

i2

OO

g

==

where pk and ik are induced by natural projections and inclusions. Then f = −g. Applying this remark toU = H1(W ), X = H1(D,V ), U1 = H1

f (W ),U2 = H1c (W ) and taking determinants we obtain the proposition.

4 Special values of p-adic L-functions

4.1 The Bloch–Kato conjecture

4.1.1 The Euler–Poincaré line (see [28], [31],[12])

Let V be a p-adic pseudo-geometric representation of Gal(Q/Q). Thus V is a finite-dimensional Qp-vectorspace equipped with a continuous action of the Galois group GS for a suitable finite set of places S containingp. Write RΓS(V ) =C•c (GS,V ) and define

RΓS,c(V ) = cone

RΓS(V )→⊕

v∈S∪∞RΓ (Qv,V )

[−1].

Fix a Zp-lattice T of V stable under the action of GS and set

∆S(V ) = det−1Qp

RΓS,c(V ), ∆S(T ) = det−1Zp

RΓS,c(T ).

Then ∆S(T ) is a Zp-lattice of the one-dimensional Qp-vector space ∆S(V ) which does not depend on thechoice of T . Therefore it defines a p-adic norm on ∆S(V ) which we denote by ‖ · ‖S. Moreover, (∆S(V ),‖ ·‖S) does not depend on the choice of S. More precisely, if Σ is a finite set of places which contains S, thenthere exists a natural isomorphism ∆S(V )

∼→ ∆Σ (V ) such that ‖ · ‖Σ = ‖ · ‖S. This allows one to define theEuler–Poincaré line ∆EP(V ) as (∆S(V ),‖ · ‖S) where S is sufficiently large. Recall that for any finite placev ∈ S we defined

RΓf (Qv,V ) =

[V Iv 1− fv−−−→V Iv

]if v 6= p[

Dcris(V )(pr,1−ϕ)−−−−−→ tV (Qp)⊕Dcris(V )

]if v = p.

At v = ∞ we set RΓf (R,V ) = [V+→ 0] , where the first term is placed in degree 0. Thus RΓf (R,V )∼→

RΓ (R,V ). For any v we have a canonical morphism locv : RΓf (Qv,V )→ RΓ (Qv,V ) which can be viewedas a local condition in the sense of [23]. Consider the diagram

78 Denis Benois

RΓS(V ) // ⊕v∈S∪∞

RΓ (Qv,V )

⊕v∈S∪∞

RΓf (Qv,V )

OO

and define

RΓf (V ) = cone

RΓS(V )⊕ ⊕

v∈S∪∞RΓf (Qv,V )

→ ⊕v∈S∪∞

RΓ (Qv,V )

[−1].

Thus, we have a distinguished triangle

RΓf (V )→ RΓS(V )⊕ ⊕

v∈S∪∞RΓf (Qv,V )

→ ⊕v∈S∪∞

RΓ (Qv,V ). (8)

Set∆ f (V ) = det−1

QpRΓf (V )⊗det−1

QptV (Qp)⊗detQpV

+.

It is easy to see that RΓf (V ) and ∆ f (V ) do not depend on the choice of S. Consider the distinguished triangle

RΓS,c(V )→ RΓf (V )→⊕

v∈S∪∞RΓf (Qv,V ).

The identity map id : Dcris(V )→ Dcris(V ) induces an isomorphism

detQpRΓf (Qp,V )' det−1Qp

tV (Qp). (9)

For v 6= p the identity map id : V Iv →V Iv gives a trivialisation

detQpRΓf (Qv,V )'Qp. (10)

Since detQpRΓf (R,V ) = detQpV+ tautologically, we obtain canonical isomorphisms

∆ f (V )' det−1Qp

RΓS,c(V )' ∆EP(V ). (11)

The cohomology of RΓf (V ) is as follows:

R0Γf (V ) = H0

S (V ), R1Γf (V ) = H1

f (V ), R2Γf (V )' H1

f (V∗(1))∗,

R3Γf (V ) = coker

(H2

S (V )→⊕v∈S

H2(Qv,V )

)' H0

S (V∗(1))∗. (12)

These groups sit in the following exact sequence:

0→ R1Γf (V )→ H1

S (V )→⊕v∈S

H1(Qv,V )

H1f (Qv,V )

→ R2Γf (V )→

H2S (V )→

⊕v∈S

H2(Qv,V )→ R3Γf (V )→ 0.

The L-function of V is defined as the Euler product

L(V,s) = ∏v

Ev(V,(Nv)−s)−1

where


Ev(V, t) =

det(1− fvt |V Iv

), if v 6= p

det(1−ϕt |Dcris(V )) if v = p.

4.1.2 Canonical trivialisations

In this paper we treat motives in the formal sense and assume all conjectures about the category of mixedmotives MM over Q which are necessary to state the Bloch–Kato conjecture (see [28], [31]). Let M be apure motive over Q and let MB and MdR denote its Betti and de Rham realisations respectively. Fix an oddprime p and denote by V = Mp the p-adic realisation of M. Then one has comparison isomorphisms

MB⊗QC ∼→MdR⊗QC, (13)

MB⊗QQp∼→V. (14)

The isomorphism (14) induces a trivialisation

Ω(et,p)M : detQpV ⊗det−1

Q MB→Qp (15)

The complex conjugation acts compatibly on MB and V and decomposes the last isomorphism into ± partswhich we denote again by Ω

(et,p)M to simplify notation

Ω(et,p)M : detQpV±⊗det−1

Q M±B →Qp. (16)

The restriction of V to the decomposition group at p is a de Rham representation and DdR(V )'MdR⊗QQp.The comparison isomorphism

V ⊗BdR∼→ DdR(V )⊗BdR (17)

induces a mapΩ

(H,p)M : detQpV ⊗det−1

QpDdR(V )→ BdR.

It is not difficult to see that there exists a finite extension L of Qurp such that Im(Ω

(H,p)M ) ⊂ LttH (V ) and we

defineΩ

(H,p)M : detQpV ⊗det−1

QpDdR(V )→ L (18)

by Ω(H,p)M = t−tH (V )Ω

(H,p)M . We remark that if V is crystalline at p then one can take L = Qur

p (see [55],Appendice C.2).

Assume that the groups H i(M) = ExtiMM(Q(0),M) are well defined and vanish for i 6= 0,1. It shouldbe possible to define a Q-subspace H1

f (M) of H1(M) consisting of "integral" classes of extensions whichis expected to be finite dimensional. It is convenient to set H0

f (M) = H0(M). The conjectures of Tate andJannsen predict that the regulator map induces isomorphisms

H if (M)⊗QQp ' H i

f (V ), i = 0,1. (19)

In this paper M will always denote a motive satisfying the following conditions

M1) M is pure of weight w6−2.M2) The p-adic realisation V of M is crystalline at p.M3) M has no subquotients isomorphic to Q(1).

These conditions imply that H0(M) = H0(M∗(1)) = 0 and H1(M∗(1)) = 0 by the weight argument.Furthermore, (19) implies that the representation V should satisfy the conditions C1,2,4) of Sect. 3.1.2.Also, from (12) it follows that

detQpRΓf (V )∼→ det−1

QpH1

f (V ). (20)

The semisimplicity of ϕ is a well known conjecture which is actually known for abelian varieties. FinallyC5) should follow from the injectivity of the syntomic regulator.

80 Denis Benois

The comparison isomorphism (13) induces an injective map

αM : M+B ⊗QR→ tM(R)

and the six-term exact sequence of Fontaine and Perrin-Riou (see [28], Sect. 6.10) degenerates into anisomorphism (the regulator map)

rM,∞ : H1f (M)⊗QR ∼→ coker(αM).

The maps αM and rM,∞ define a map

RM,∞ : det−1Q tM(Q)⊗detQM+

B ⊗detQH1f (M)→ R

Fix bases ω f ∈ detQH1f (M), ωtM ∈ detQtM(Q) and ω

+MB∈ detQM+

B . Set ωM = (ω f ,ωtM ,ω+MB

) and define

RM,∞(ωM) = RM,∞(ω−1tM ⊗ω

+MB⊗ω f ).

Using (11), (16), (20) and the isomorphisms (19) define

iωM ,p : ∆EP(V )' det−1Qp

tV (Qp)⊗detQpV+⊗detQpH1f (V )→Qp (21)

by x = iωM ,p(x)(ω−1tM ⊗ω

+MB⊗ω f ).

Consider now the case of the dual motive M∗(1). Again one has ∆EP(V ∗(1))' ∆ f (V ∗(1)) where

∆ f (V ∗(1))' det−1Qp

tV ∗(1)(Qp)⊗detQpV ∗(1)+⊗detQpH1f (V ).

The map αM∗(1) : M∗(1)+B ⊗Q R→ tM∗(1)(R) is surjective and it is related to αM by the canonical dual-ity coker(αM)× ker(αM∗(1))→ R (see [28], Sect. 5.4). The six-term exact sequence degenerates into anisomorphism

rM∗(1),∞ : H1f (M)∗⊗QR' ker(αM∗(1)).

This allows one to define a map

RM∗(1),∞ : det−1Q tM∗(1)(Q)⊗detQM∗(1)+B ⊗detQH1

f (M)→ R.

We fix bases ωtM∗(1) ∈ detQtM∗(1)(Q) and ω+M∗(1)B

∈ detQM∗(1)+B and set

ωM∗(1) = (ωtM∗(1) ,ω+M∗(1)B

,ω f ),

RM∗(1),∞(ωM∗(1)) = RM∗(1),∞(ω−1tM∗(1)

⊗ω+M∗(1)B

⊗ω f ).

Again this data defines a trivialisation

iωM∗(1),p : ∆EP(V ∗(1))→Qp. (22)

It is conjectured that the L-functions L(V,s) and L(V ∗(1),s) are well defined complex functions, have mero-morphic continuation to the whole of C and satisfy some explicit functional equation (see [31], Chap. III).One expects that they do not depend on the choice of the prime p and we will denote them by L(M,s) andL(M∗(1),s) respectively. The conjectures about special values of these functions can be stated as follows.

Conjecture 2 (Beilinson–Deligne). The L-function L(M,s) does not vanish at s = 0 and

L(V,0)RM,∞(ωM)

∈Q∗.

The L-function L(M∗(1),s) has a zero of order r = dimQp H1f (M) at s= 0. Let L(M∗(1),0)= lims→0 s−rL(M∗(1),s).

Then


L(M∗(1),0)RM∗(1),∞(ωM∗(1))

∈Q∗.

Conjecture 3 (Bloch–Kato). Let T be a Zp-lattice of V stable under the action of GS. Then

iωM ,p(∆EP(T )) =L(M,0)

RM,∞(ωM)Zp,

iωM∗(1),p(∆EP(T ∗(1))) =L(M∗(1),0)

RM∗(1),∞(ωM∗(1))Zp.

4.1.3 Compatibility with functional equation

The compatibility of the Bloch–Kato conjecture with the functional equation follows from the conjectureCEP(V ) of Fontaine and Perrin-Riou about local Tamagawa numbers (see [31], Chap. III, Sect. 4.5.4). Moreprecisely, define

Γ∗(V ) = ∏

i∈ZΓ∗(−i)hi(V ) (23)

where hi(V ) = dimQp (gri(DdR(V ))) and

Γ∗(i) =

(i−1)! if i > 0,(−1)i

(−i)! if i6 0.

The exact sequence0→ tV ∗(1)(Qp)

∗→ DdR(V )→ tV (Qp)→ 0

allows to consider ωMdR = ωtM ⊗ω−1tM∗(1) ∈ detQpDdR(V ). Choose bases ω

+T ∈ detZpT+ and ω

−T ∈ detZpT−

and set ωT =ω+T ⊗ω

−T ∈ detZpT and ω

+T ∗(1)=(ω−T )∗ ∈ detZpT ∗(1)+. Define p-adic periods Ω

(et,p)M (ω+

T ,ω+MB

)

and Ω(H,p)M (ωT ,ωMdR) by ω

+T = Ω

(et,p)M (ω+

T ,ω+MB

)ω+MB

and ωT = Ω(H,p)M (ωT ,ωMdR)ωMdR using isomor-

phisms (15) and (18). Then the conjecture CEP(V ) implies that

iωM∗(1),p(∆EP(T ∗(1)))

Ω(et,p)M∗(1)(ω

+T ∗(1),ω

+M∗(1)B

)= Γ

∗(V ) Ω(H,p)M (ωT ,ωMdR)

iωM ,p(∆EP(T ))

Ω(et,p)M (ω+

T ,ω+MB

)(24)

(see [55], Appendice C). We remark that for crystalline representations CEP(V ) is proved in [BB12].


4.2.1 p-adic Beilinson conjecture

We maintain previous notation and conventions. Let M be a motive which satisfies the conditions M1-3) ofSect. 4.1.2 and let V denote the p-adic realisation of M. We fix bases ω

+MB∈ detQM+

B , ωtM ∈ detQtM(Q) andω f ∈ detQH1

f (M). We also fix a lattice T in V stable under the action of GS and a basis ω+T ∈ detZpT+.

To simplify notation we will assume that the choices of ω+MB

and ω+T are compatible, namely that

Ω(et,p)M (ω+

T ,ω+MB

) = 1. Let D be a regular subspace of Dcris(V ). We fix a Zp-lattice N of D and a basisωN ∈ detZpN. By the analogy with the archimedean case we can consider the p-adic regulator as a maprV,D : H1

f (V )→ coker(αV,D) whereαV,D : D→ tV (Qp)

is the natural projection. Set ωV,N = (ωtM ,ωN ,ω f ) and denote by RV,D(ωV,N) the determinant of rV,D com-puted in the bases ω f and ωtM⊗ω

−1N . Namely, RV,D(ωV,N) is the image of ω

−1tM ⊗ωN⊗ω f under the induced

isomorphism

82 Denis Benois

RV,D : det−1Qp

tV (Qp)⊗detQpD⊗detQpH1f (V )→Qp.

Now, consider the projectionαV ∗(1),D⊥ : D⊥→ tV ∗(1)(Qp).

A standard argument from linear algebra shows that αV ∗(1),D⊥ is surjective and is related to αV,D by thecanonical duality coker(αV,D)×ker(αV ∗(1),D)→Qp. This defines isomorphisms

det−1Qp

tV ∗(1)(Qp)⊗detQpD⊥ ' detQp(ker(αV ∗(1),D⊥))' det−1Qp

(coker(αV,D))

and composing this map with the determinant of rV,D we have again a trivialisation

RV ∗(1),D⊥ : det−1Qp

tV ∗(1)(Qp)⊗detQpD⊥⊗detQp H1f (V )→Qp.

Choose a lattice N⊥ ⊂ D⊥, fix bases ωtM∗(1) and ωN⊥ of detQptV ∗(1)(Qp) and detZpN⊥ respectively and setωV,N⊥ = (ωtM∗(1) ,ωN⊥ ,ω f ).

Perrin-Riou conjectured (see [55]) that there exists an analytic p-adic L-functionLp(T,N,s) which interpolates special values of the complex L-function L(M,s). In particular one expectsthat if p−1 is not an eigenvalue of ϕ acting on D then Lp(T,N,s) does not vanish at s = 0 and

Lp(T,N,0)RV,D(ωV,N)

= E(V,D)L(M,0)

RM,∞(ωM)

where

E(V,D) = det(1− p−1ϕ−1 | D) det(1− p−1

ϕ−1 | D⊥) =

= det(1− p−1ϕ−1 | D) det(1−ϕ | Dcris(V )/D).

Dually it is conjectured that there exists a p-adic L-function Lp(T ∗(1),N⊥,s) which interpolates specialvalues of L(M∗(1),s). One expects that if 1 is not an eigenvalue of ϕ acting on the quotient Dcris(V ∗(1))/D⊥

then Lp(T ∗(1),N⊥,s) has a zero of order r = dimQ H1f (M) at s = 0 and

L∗p(T∗(1),N⊥,0)

RV ∗(1),D⊥(ωV ∗(1),N⊥)= E(V ∗(1),D⊥)

L∗(M∗(1),0)RM∗(1),∞(ωM)

.

These properties of p-adic L-functions can be viewed as p-adic analogues of Beilinson conjectures and werefer the reader to [55], Chap. 4 and [18], Sect. 2.8 for more detail. Note that from the definition it is clearthat E(V,D) = E(V ∗(1),D⊥). One can also write E(V,D) in the form

E(V,D) = Ep(V,1)det(

1− p−1ϕ−1

1−ϕ|D).

4.2.2 Extra zero conjecture

Assume now that Dϕ=p−1 6= 0. Since M is crystalline at p, this can occur only if M is of weight −2. Set

e = dimQpDϕ=p−1= dimQp(D

⊥+Dcris(V ∗(1))ϕ=1)/D⊥).

Assume that the p-adic realisation V of M satisfies the conditions C1-5) ofSect. 3.1.2. Decompose D into the direct sum D = D−1⊕Dϕ=p−1

and define

E+(V,D) = E+(V ∗(1),D⊥) = det(1− p−1ϕ−1 | D−1) det(1− p−1

ϕ−1 | D⊥). (25)

We propose the following conjecture about the behavior of p-adic L-functions at s = 0.

Conjecture 4. i) The p-adic L-function Lp(T,N,s) has a zero of order e at s = 0 and


L∗p(T,N,0)RV,D(ωV,N)

= −L (V,D)E+(V,D)L(M,0)

RM,∞(ωM).

ii) The p-adic L-function Lp(T ∗(1),N⊥,s) has a zero of order e+ r where r = dimQ H1f (M) at s = 0 and

L∗p(T∗(1),N⊥,0)

RV ∗(1),D⊥(ωV ∗(1),N⊥)= L (V,D)E+(V ∗(1),D⊥)

L∗(M∗(1),0)RM∗(1),∞(ωM∗(1))

.

Remarks 1) If H1f (M) = 0 the p-adic regulator vanishes and we recover the conjecture formulated in [3],

Sect. 2.3.2.2) The regulators RM,∞(ωM) and RV,D(ωV,N) are well defined up to a sign and in order to obtain equal-

ities in the formulation of our conjecture one should make the same choice of signs in the definitions ofRM,∞(ωM) and RV,D(ωV,N). See [55], Sect. 4.2 for more detail.

3) Our conjecture is compatible with the expected functional equation for p-adic L-functions. See Sect.2.5 of [55] and Sect. 5.2.5 below.

5 The module of p-adic L-functions

5.1 The Selmer complex

5.1.1 Iwasawa cohomology

Let Γ denote the Galois group of Q(ζp∞)/Q and Γn = Gal(Q(ζp∞)/Q(ζpn)). Set Λ = Zp[[Γ1]] and Λ(Γ ) =

Zp[∆ ]⊗Zp Λ . For any character η ∈ ∆ put

eη =1|∆ | ∑g∈∆

η−1(g)g.

Then Λ(Γ ) =⊕

η∈∆

Λ(Γ )(η) where Λ(Γ )(η) =Λeη and for any Λ(Γ )-module M one has a canonical decom-

positionM '

⊕η∈∆

M(η), M(η) = eη(M).

We write η0 for the trivial character of ∆ and identify Λ with Λ(Γ )eη0 .Let V be a p-adic pseudo-geometric representation unramified outside S. Set d(V ) = dim(V ) and

d±(V ) = dim(V c=±1). Fix a Zp-lattice T of V stable under the action of GS. Let ι : Λ(Γ )→ Λ(Γ ) de-note the canonical involution g 7→ g−1. Recall that the induced module IndQ(ζp∞ )/Q(T ) is isomorphic to(Λ(Γ )⊗Zp T )ι (see [23], Sect. 8.1). Define

H iIw,S(T ) = H i

S((Λ(Γ )⊗Zp T )ι),

H iIw(Qv,T ) = H i(Qv,(Λ(Γ )⊗Zp T )ι) for any finite place v.

From Shapiro’s lemma it follows immediately that

H iIw,S(T ) = lim←−

coresH i

S(Q(ζpn),T ), H iIw(Qp,T ) = lim←−

coresH i(Qp(ζpn),T ).

Set H iIw,S(V ) = H i

Iw,S(T )⊗Zp Qp and H iIw(Qv,V ) = H i

Iw(Qv,T )⊗Zp Qp. In [55] Perrin-Riou proved thefollowing results about the structure of these modules.

i) H iIw,S(V ) = 0 and H i

Iw(Qv,T ) = 0 if i 6= 1,2;

84 Denis Benois

ii) If v 6= p, then for each η ∈ ∆ the η-component H iIw(Qv,T )(η) is a finitely-generated Λ -torsion mod-

ule. In particular, H1Iw(Qv,T )' H1(Qur

v /Qv,(Λ(Γ )⊗Zp T Iv)ι).

iii) If v = p then H2Iw(Qp,T )(η) are finitely-generated Λ -torsion modules. Moreover, for each η ∈ ∆

rankΛ

(H1

Iw(Qp,T )(η))= d, H1

Iw(Qp,T )(η)tor ' H0(Qp(ζp∞) ,T )(η).

Remark that by local duality H2Iw(Qp,T )' H0(Qp(ζp∞),V ∗(1)/T ∗(1)).

iv) If the weak Leopoldt conjecture holds for the pair (V,η), i.e. ifH2

S (Q(ζp∞),V/T )(η) = 0, then H2Iw,S(T )

(η) is Λ -torsion and

rankΛ

(H1

Iw,S(T )(η))=

d−(V ), if η(c) = 1d+(V ), if η(c) =−1.

Passing to the projective limit in the Poitou-Tate exact sequence one obtains an exact sequence

0→ H2S (Q(ζp∞),V ∗(1)/T ∗(1))∧→ H1

Iw,S(T )→⊕v∈S

H1Iw(Qv,T )→

→ H1S (Q(ζp∞),V ∗(1)/T ∗(1))∧→ H2

Iw,S(T )→

→⊕v∈S

H2Iw(Qv,T )→ H0

S (Q(ζp∞),V ∗(1)/T ∗(1))∧→ 0. (26)

Define

RΓIw,S(T ) = C•c (GS,(Λ(Γ )⊗Zp T )ι),

RΓIw(Qv,T ) = C•c (Gv,(Λ(Γ )⊗Zp T )ι),

RΓS(Q(ζp∞),V ∗(1)/T ∗(1)) = C•c (GS,HomZp(Λ(Γ ),V ∗(1)/T ∗(1))).

Then the sequence (26) is induced by the distinguished triangle

RΓIw,S(T )→⊕v∈S

RΓIw(Qv,T )→ (RΓS(Q(ζp∞),V ∗(1)/T ∗(1))ι)∧[−2]

(see [23], Theorem 8.5.6). Finally, we have the usual descent formulas

RΓIw,S(T )⊗LΛ Zp ' RΓS(T ), RΓIw(Qv,T )⊗L

Λ Zp ' RΓ (Qv,T )

(see [23], Proposition 8.4.22).

5.1.2 The complex RΓ(η0)

Iw,h (D,V )

For the remainder of this chapter we assume that V satisfies the conditions C1-5) of Sect. 3.1.2 and thatthe weak Leopoldt conjecture holds for (V,η0) and (V ∗(1),η0). We remark that these assumptions arenot independent. Namely, by [55], Proposition B.5 C4) and C5) imply the weak Leopoldt conjecture for(V ∗(1),η0). From the same result it follows that the vanishing of H1

f (V∗(1)) implies the weak Leopoldt

conjecture for (V,η0) if in addition we assume that H0(Qp,V ∗(1)) = 0.To simplify notations we write H for H(Γ1). Fix a regular subspace D of Dcris(V ) and a Zp-lattice N of

D. Set Dp(N,T )(η0) = N⊗Zp Zp[[X ]]ψ=0 ' N⊗Zp Λ , and define

RΓ(η0)

Iw (Qp,N,T ) =Dp(N,T )(η0)[−1],

RΓ(η0)

Iw (Qp,D,V ) = RΓ(η0)

Iw (Qp,N,T )⊗Zp Qp.

Consider Perrin-Riou’s exponential map


ExpεV,h : RΓ

(η0)Iw (Qp,N,T )⊗Λ H→ RΓ

(η0)Iw (Qp,T )⊗L

Λ H

which will be viewed as a local condition at p. If v 6= p the inertia group Iv acts trivially on Λ . Set

RΓ(η0)

Iw (Qv,N,T ) =[T Iv ⊗Λ

ι 1− fv−−−→ T Iv ⊗Λι

]where the first term is placed in degree 0. We have a commutative diagram

RΓ(η0)

Iw,S (T )⊗Λ H // ⊕v∈S

RΓ(η0)

Iw (Qv,T )⊗Λ H

(⊕v∈S

RΓ(η0)

Iw (Qv,N,T ))⊗Λ H .

OO(27)

Consider the associated Selmer complex

RΓ(η0)

Iw,h (D,V ) = cone

[(RΓ

(η0)Iw,S (T )

⊕(⊕v∈S

RΓ(η0)

Iw (Qv,N,T )

))⊗Λ H→

⊕v∈S

RΓ(η0)

Iw (Qv,T )⊗Λ H

][−1]

It is easy to see that it does not depend on the choice of S. Our main result about this complex is thefollowing theorem.

Theorem 4. Assume that V satisfies the conditions C1-5) and that the weak Leopoldt conjecture holds for(V,η0) and (V ∗(1),η0). Let D be a regular subspace of Dcris(V ). Assume that L (V,D) 6= 0. Then

i) RiΓ(η0)

Iw,h (D,V ) are H-torsion modules for all i.

ii) RiΓ(η0)

Iw,h (D,V ) = 0 for i 6= 2,3 and

R3Γ

(η0)Iw,h (D,V )'

(H0(Q(ζp∞),V ∗(1))∗

)(η0)⊗Λ H.

iii) The complex RΓ(η0)

Iw,h (D,V ) is semisimple in the sense that for each i the natural map

RiΓ

(η0)Iw,h (D,V )Γ → Ri

Γ(η0)

Iw,h (D,V )Γ

is an isomorphism.

5.1.3 Proof of Theorem 5.1.3.

We leave the proof of the following lemma as an easy exercise.

Lemma 2. Let A and B be two submodules of a finitely-generated free H-module M. Assume that thenatural maps AΓ1 →MΓ1 and BΓ1 →MΓ1 are both injective. Then AΓ1 ∩BΓ1 = 0 implies that A∩B = 0.

5.1.4.2. Since H0Iw,S(V ) and H0

Iw(Qv,V ) are zero, we have R0Γ(η0)

Iw,h (D,V )= 0. Next, by definition R1Γ(η0)

Iw,h (D,V ) =

ker( f ) where

f :

H1Iw,S(T )

(η0)⊕

Dp(N,T )(η0)⊕

v∈S−pH1

Iw, f (Qv,T )(η0)

⊗H→

⊕v∈S

H1Iw(Qv,T )(η0)⊗H

86 Denis Benois

is the map induced by (27). If v ∈ S−p one has

H1Iw, f (Qv,T )(η0) = H1

Iw(Qv,T )(η0) = H1(Qurv /Qv,(Λ ⊗T Iv)ι).

ThusR1

Γ(η0)

Iw,h (D,V ) =(

H1Iw,S(T )

(η0)⊗Λ H)∩(

ExpεV,h

(Dp(D,T )(η0)

)⊗Λ H

)in H1

Iw(Qp,T )(η0)⊗Λ H. Put

A = ExpεV,h(D−1⊗H)⊕X−1Expε

V,h(Dϕ=p−1 ⊗H)⊂ H1

Iw(Qp,T )(η0)⊗Λ H,

where we identify X with the operator γ1−1 (see Sect. 2.2.1). By Theorem 3 and Proposition 6 AΓ1 injects

into H1(Qp,V ). Since T HQp is the torsion submodule of H1Iw(Qp,T ), the H-module M =

(H1

Iw(Qp,T )

T HQp

)(η0)

⊗Λ

H is free and A →M. Since T GQp = 0 one has MΓ1 = H1Iw(Qp,V )Γ ⊂ H1(Qp,V ) and we obtain that AΓ1

injects into MΓ1 .

Set B=

(H1

Iw,S(T )

T HQ

)(η0)

⊗Λ H. The weak Leopoldt conjecture for (V ∗(1),η0) together with the fact that

H1Iw(Qv,T ) are Λ -torsion for v ∈ S−p imply that B →M. Since the image of H1

Iw(Qv,V )Γ in H1(Qv,V )is contained in H1

f (Qv,V ), the image of H1Iw,S(V )Γ in H1

S (V ) is in fact contained in H1f ,p(V ). From C5) it

follows that H1f ,p(V ) injects into H1(Qp,V ) and we have

H1Iw,S(V )

(η0)Γ1

= H1Iw,S(V )Γ → H1

f ,p(V ) → H1(Qp,V ).

Thus BΓ1 ⊂MΓ1 . We shall prove that R1Γ(η0)

Iw,h (D,V ) = 0. By Lemma 2 it suffices to show that AΓ1 ∩BΓ1 =

0. Now we claim that AΓ1 ∩H1f ,p(V ) = 0. First note that by Lemma 1

H1f ,p(V ) →

H1(Qp,V )

H1(D−1).

On the other hand, from Theorem 3 it follows that

ExpεV,h(D−1⊗H)Γ1 = expV,Qp

(D−1)⊂ H1(D−1).

Now Proposition 6 implies that the image of AΓ1 inH1(Qp,V )

H1(D−1)coincides with H1

c (W ). But L (V,D) 6= 0

if and only if H1D(V )∩H1

c (W ) = 0 where H1D(V ) denotes the inverse image of H1(W ) in H1

f ,p(V ) (see

Lemma 1 ii)). This proves the claim and implies that R1Γ(η0)

Iw,h (D,V ) = 0.

5.1.4.3. We shall show that R2Γ(η0)

Iw,h (D,V ) is H-torsion. By definition, we have an exact sequence

0→ coker( f )→ R2Γ

(η0)Iw,h (D,V )→ III2

Iw,S(V )(η0)⊗ΛQpH→ 0, (28)

where

III2Iw,S(V ) = ker

(H2

Iw,S(V )→⊕v∈S

H2Iw(Qv,V )

).

It follows from the weak Leopoldt conjecture that III2Iw,S(V ) is ΛQp -torsion. On the other hand, as H is a

Bézout ring (see [46]), the formulas

rankΛ H1Iw,S(T )

(η0) = d−(V ),

rankΛ H1Iw(Qp,T )(η0) = d(V ),

rankΛDp(N,T ) = d+(V )


together with the fact that R1Γ(η0)

Iw,h (D,V ) = 0 imply that coker( f ) is H-torsion. We have therefore provedthat R2ΓIw,h(D,V ) is H-torsion. Finally, the Poitou–Tate exact sequence gives that

R3Γ

(η0)Iw,h (D,V ) =

(H0(Q(ζp∞),V ∗(1))∗

)(η0)⊗ΛQpH

is also H-torsion.5.1.4.4. Now we prove the semisimplicity of RΓ

(η0)Iw,h (D,V ). First write

H1Iw,S(V )(η0) 'Λ

d−(V )Qp

⊕H1Iw,S(V )

(η0)tor .

Since H1Iw,S(V )tor⊂H1

Iw(Qp,V )tor =V HQp , (H1Iw,S(V )tor)Γ = 0 by the snake lemma. Thus dimQp H1

Iw,S(V )(η0)Γ1

=

d−(V ). On the other hand dimQp H1f ,p(V ) = d−(V )+dimQp H0(Qp,V ∗(1)) by (6) and the dimension ar-

gument shows that in the commutative diagram

0 // H1Iw,S(V )

(η0)Γ1

//

H1f ,p(V ) //

H0(Qp,V ∗(1))∗ //

0

0 // H1Iw(Qp,V )

(η0)Γ1

// H1(Qp,V ) // H0(Qp,V ∗(1))∗ // 0

(29)

with obviously exact upper line the bottom line is also exact. This implies immediately that the natural map

H1Iw(Qp,V )

(η0)Γ1

H1Iw,S(V )

(η0)Γ1

+H1(D−1)−→

H1(Qp,V )

H1f ,p(V )+H1(D−1)

is an isomorphism.Consider the exact sequence

0→(

H1Iw,S(T )

(η0)⊕Dp(N,T )(η0))⊗H→ H1

Iw(Qp,T )(η0)⊗H→ coker( f )→ 0.

Recall that ExpεV,h,0 : D→H1

Iw(Qp,V )Γ denotes the homomorphism induced by the large exponential map.Applying the snake lemma, and taking into account that Im(Expε

V,h,0) = expV,Qp(D−1) = H1(D−1) and

ker(ExpεV,h,0) = Dϕ=p−1

(see for example [BB12], Propositions 4.17 and 4.18 or the proof of Proposition 6)we obtain (by regularity of D)

coker( f )Γ1 = ker(

H1Iw,S(V )

(η0)Γ1⊕D

ExpεV,h,0−−−−→ H1(Qp,V )

)= Dϕ=p−1

coker( f )Γ1 =H1

Iw(Qp,V )(η0)Γ1

H1Iw,S(V )

(η0)Γ1

+H1(D−1)=

H1(Qp,V )

H1f ,p(V )+H1(D−1)

. (30)

Thus, one has a commutative diagram

coker( f )Γ1 //

Dϕ=p−1

δD,h

coker( f )Γ1// H1(Qp,V )

H1f ,p(V )+H1(D−1)

.

(31)

where horizontal arrows are isomorphisms, the left vertical arrow is the natural projection and the rightvertical row is the map defined in Sect. 3.2.2. From Proposition 7 it follows that coker( f )Γ1 → coker( f )Γ1is an isomorphism if and only if L (V,D) 6= 0.

88 Denis Benois

On the other hand, the arguments of [55], Sect. 3.3.4 show that III2Iw,S(V )Γ = III2

Iw,S(V )Γ = 0. We re-mark that Perrin-Riou assumes thatDcris(V )ϕ=1 = Dcris(V )ϕ=p−1

= 0, but her proof works in our case without modifications and we repeatit for the convenience of the reader. Consider the commutative diagram (where we write III2

Iw(V ) insteadIII2

Iw,S(V ) to abbreviate notation)

0

0

H1

Iw(V )Γ// //

⊕v∈S

H1Iw(Qv,V )Γ

//

H1S (Q(ζp∞ ),V ∗(1))∗

Γ// //

=

III2Iw(V )Γ

H1f ,p(V ) // //

⊕v∈Sv 6=p

H1f (Qv,V )

⊕H1(Qp,V ) // //

H1S (V

∗(1))∗

H0(Qp,V ∗(1))∗

= // H0(Qp,V ∗(1))∗

0 0.

The top row of this diagram is obtained by taking coinvariants in the Poitou–Tate exact sequence. Thus itis exact. The middle row is obtained from the exact sequence

0→ H1S (V

∗(1))→ H1(Qp,V ∗(1))⊕ ⊕

v∈S−p

H1(Qv,V ∗(1))H1

f (Qv,V ∗(1))→ H1

f ,p(V )∗→ 0

by taking duals. In particular, in the first and second lines of the diagram the first arrows are injections.Here we use the condition H1

f (V∗(1)) = 0. The exactness of the left and middle columns follows from the

diagram (29). The isomorphism from the right column comes from the exact sequence

0→ H1(Γ ,H0S (Q(ζp∞),V ∗(1)))→ H1

S (V∗(1))→ H1

S (Q(ζp∞),V ∗(1))Γ → 0

together with the remark that H1(Γ ,H0S (Q(ζp∞),V ∗(1))) = 0 because

H0S (Q(ζp∞),V ∗(1))Γ =H0

S (Q,V ∗(1))) = 0 by C2). Now an easy diagram chase shows that III2Iw,S(V )Γ = 0.

Finally, from dimQpIII2Iw,S(V )Γ 6 dimQpIII2

Iw,S(V )Γ it follows that III2Iw,S(V )Γ = 0. Therefore, applying

the snake lemma to (28) we obtain a commutative diagram

coker( f )Γ1 //

R2Γ(η0)

Iw,h (D,V )Γ1

coker( f )Γ1// R2Γ

(η0)Iw,h (D,V )Γ1 ,

(32)

where the horizontal arrows are isomorphisms and the vertical arrows are natural projections. This provesthat RΓ

(η0)Iw,h (D,V ) is semisimple in degree 2. The semisimplicity in degree 3 is obvious because by ii)

R3Γ(η0)

Iw,h (D,V )Γ1 = R3Γ(η0)

Iw,h (D,V )Γ1 = 0. This completes the proof of Theorem 4. ut

This following corollary relates the projection map

R2Γ

(η0)Iw,h (D,V )Γ1 −→ R2

Γ(η0)

Iw,h (D,V )Γ1

to the map λD defined in Proposition 7 and therefore to the L -invariant. This relation plays the key role inthe proof of Theorem 5 below (Theorem 2 of Introduction).


Corollary 1. i) One has canonical isomorphisms

R2Γ

(η0)Iw,h (D,V )Γ1 ∼→ Dϕ=p−1

,

R2Γ

(η0)Iw,h (D,V )Γ1

∼→H1(Qp,V )

H1f ,p(V )+H1(D−1)

.

ii) The exponential map induces an isomorphism of Dϕ=p−1onto R2Γ

(η0)Iw,h (D,V )Γ1 and the following

diagram commutes

Dϕ=p−1 ∼ //

λD

R2Γ(η0)

Iw,h (D,V )Γ1

Dϕ=p−1 (h−1)!expV // R2Γ(η0)

Iw,h (D,V )Γ1

where the map λD is defined in Proposition 7.

Proof. The corollary follows from diagrams (30), (31), (32) and the definition of λD.

5.2 The module of p-adic L-functions

5.2.1 The canonical trivialisation

We preserve the notation and conventions of Sect. 4.2. Let D be a regular subspace of Dcris(V ) and as-sume that L (V,D) 6= 0. We review Perrin-Riou’s definition of the module of p-adic L-functions using theformalism of Selmer complexes. Set

∆Iw,h(D,V ) =

det−1ΛQp

(RΓ

(η0)Iw,S (V )⊕

(⊕

v∈SRΓ

(η0)Iw (Qv,D,V )

))⊗detΛQp

(⊕

v∈SRΓ

(η0)Iw (Qv,V )

).

The distinguished triangle

RΓ(η0)

Iw,S (D,V )→

(RΓ

(η0)Iw,S (V )

⊕(⊕v∈S

RΓ(η0)

Iw (Qv,D,V )

))⊗H→

→

(⊕v∈S

RΓ(η0)

Iw (Qv,V )

)⊗H→ RΓ

(η0)Iw,S (D,V )[1]

gives an isomorphism ∆Iw,h(D,V )⊗ΛQpH ' det−1

H RΓ(η0)

Iw,S (D,V ). Let K denote the field of fractions of H.

By Theorem 4, all RiΓ(η0)

Iw,S (D,V ) are H-torsion and we have a canonical map

det−1H RΓ

(η0)Iw,S (D,V )' ⊗

i∈2,3det(−1)i+1

H RiΓ

(η0)Iw,S (D,V ) →K.

The composition of these maps gives a trivialization

iV,Iw,h : ∆Iw,h(D,V )−→K.

90 Denis Benois

5.2.2 Local conditions

We compare local conditions coming from Perrin-Riou’s theory to those of Bloch–Kato. Roughly speakingthe computations of this section explain the appearance of the Euler-like factor E+(V,D) in the formula forthe special value of LIw,h(T,N,s). They will be used in the proof of Theorem 5.

Set RΓ (Qp,D,V ) = D[−1] and define

S = cone(

1− p−1ϕ−1

1−ϕ: RΓ (Qp,D,V )→ RΓf (Qp,V )

)[−1]. (33)

Since detQp RΓf (Qp,V ) is canonically isomorphic to det−1Qp

tV (Qp) by (9), the distinguished triangle

S→ RΓ (Qp,D,V )→ RΓf (Qp,V )→ S[1]

induces an isomorphismαS : detQptV (Qp)

∼→ detQpD⊗detQpS. (34)

Explicitly

S = [D⊕Dcris(V )→ Dcris(V )⊕ tV (Qp)] [−1]'[D⊕Dcris(V )→ Dcris(V )⊕D)] [−1],

where the unique non-trivial map is given by

(x,y) 7→((1−ϕ)y,

(1− p−1ϕ−1

1−ϕx+ y

)(mod Fil0Dcris(V ))

).

Thus

H1(S) = Dϕ=p−1, H2(S) =

tV (Qp)

(1− p−1ϕ−1)D' Dcris(V )

Fil0Dcris(V )+D−1. (35)

From the semi-simplicity of1− p−1ϕ−1

1−ϕit follows that the natural projection H1(S)⊕H1

f (V )→ H2(S) is

an isomorphism and we have a canonical trivialization

βS : detQpS⊗detQpRΓf (V )'' det−1

QpH1(S)⊗detQpH2(S)⊗det−1

QpH1

f (V )'Qp. (36)

The composition of βS with αS gives an isomorphism

θS : detQptV (Qp)⊗detQpRΓf (V )αS'

detQpD⊗detQpS⊗detQpRΓf (V )id⊗βS' detQpD. (37)

Fix bases ωtV ∈ detQp tV (Qp), ωD ∈ detQp D and ω f ∈ detQp H1f (V ). Let RV,D(ωV,D) denote the determinant

of the regulator maprV,D : H1

f (V )→ Dcris(V )/(Fil0Dcris(V )+D)

defined in Sect. 3.1.3 with respect to ω f and ωtV ⊗ω−1D .

Lemma 3. i) Let f : W →W be a semi-simple endomorphism of a finite-dimensional k-vector space W.The canonical projection ker( f )→ coker( f ) is an isomorphism and the tautological exact sequence

0→ ker( f )→Wf−→W → coker( f )→ 0

induces an isomorphism

det∗ f : detk(W )→ detk(W )⊗detk(ker( f ))⊗det−1k (coker( f ))→ detk(W ).


Furthermore det∗ f (x) = det( f |coker( f )).ii) The map θS sends ωtV ⊗ω

−1f onto

det∗(

1− p−1ϕ−1

1−ϕ|D)−1

Ep(V,1)−1RV,D(ωV,D)−1

ωD

Proof. The proof is straightforward and is omitted here.

5.2.3 Definition of the module of p-adic L-functions

In this subsection we interpret Perrin-Riou’s construction of the module of p-adic L-functions in termsof [23]. Fix a Zp-lattice N of D and set

∆Iw,h(N,T ) =

det−1Λ

(RΓ

(η0)Iw,S (T )

⊕(⊕v∈S

RΓ(η0)

Iw (Qv,N,T )

))⊗detΛ

(⊕v∈S

RΓ(η0)

Iw (Qv,T )

).

The module of p-adic L-functions associated to (N,T ) is defined as

L(η0)Iw,h(N,T ) = iV,Iw,h

(∆Iw,h(N,T )

)⊂K.

Fix a generator f (γ1−1) of L(η0)Iw,h(N,T ) and define a meromorphic p-adic function

LIw,h(T,N,s) = f (χ(γ1)s−1).

Let now V be the p-adic realisation of a pure motive M over Q which satisfies the conditions M1-3) ofSect. 4.1.2. As we saw in Sect. 4.1.2 one expects that V satisfies C1-5). We fix bases ω f ∈ detQH1

f (M),

ωtM ∈ detQtM(Q) and use the same notation for their images in detQpH1f (V ) and detQptV (Qp) respec-

tively. As in Sect. 4.1.3 choose bases ω+MB∈ detQM+

B and ω+T ∈ detZpT+ and define the p-adic period

Ω(et,p)M (ω+

T ,ω+MB

) ∈ Qp by ω+T = Ω

(et,p)M (ω+

T ,ω+MB

)ω+MB

using the comparison isomorphism (13) (see also(14)). Let ωN be a generator of detZpN.

Theorem 5. Assume that V satisfies C1-5) and that the weak Leopoldt conjecture holds for (V,η0) and(V ∗(1),η0). Let D be a regular submodule of Dcris(V ). Assume that L (V,D) 6= 0. Then

i) LIw,h(T,N,s) is a meromorphic p-adic function which has a zero at s = 0 of order e = dimQp(Dϕ=p−1

).ii) Let L∗Iw,h(T,N,0) = lims→0 s−eLIw,h(T,N,s) be the special value of

LIw,h(T,N,s) at s = 0. Then

L∗Iw,h(T,N,0)

RV,D(ωV,N)∼p Γ (h)d+(V )L (V,D)E+(V,D)

iωM ,p (∆EP(T ))

Ω(et,p)M (ω+

T ,ω+MB

),

where iωM ,p and E+(V,D) are defined by (21) and (25) respectively and Γ (h) = (h−1)!.

5.2.4 Proof of Theorem 5.2.5

5.2.6.1. First recall the formalism of Iwasawa descent which will be used in the proof. The result we needis proved in [13]. This is a particular case of Nekovár’s descent theory [23]. Let C• be a perfect complex ofH-modules and let C•0 =C•⊗L

HQp. We have a natural distinguished triangle

C• X−→C•→C•0 →C•[1],

where X = γ1−1. In each degree this triangle gives a short exact sequence

92 Denis Benois

0→ Hn(C•)Γ1 → Hn(C•0)→ Hn+1(C•)Γ1 → 0.

One says that C• is semisimple if the natural map

Hn(C•)Γ1 → Hn(C•)→ Hn(C•)Γ1 (38)

is an isomorphism in all degrees. If C• is semisimple, there exists a canonical trivialisation of detQpC•0 ,namely

ϑ : detQpC•0 ' ⊗n∈Z

det(−1)n

QpHn(C0)'

' ⊗n∈Z

(det(−1)n

QpHn(C•)Γ1 ⊗det(−1)n

QpHn+1(C•)Γ1

)'

' ⊗n∈Z

(det(−1)n

QpHn(C•)Γ1 ⊗det(−1)n−1

QpHn(C•)Γ1

)'Qp

where the last map is induced by (38). We now suppose that C•⊗HK is acyclic and write ϑ∞ : detHC•→K

for the associated morphism. Then ϑ∞(detHC•) = fH, where f ∈K. Let r be the unique integer such thatX−r f is a unit of the localization H0 of H with respect to the principal ideal XH.

Lemma 4. Assume that C•⊗K is acyclic and C• is semisimple. Then

r = ∑n∈Z

(−1)n+1 dimQp Hn(C•)Γ1

and there exists a commutative diagram

detHC•X−rϑ∞ //

⊗LHQp

H0

detQpC•0

ϑ // Qp

in which the right vertical arrow is the augmentation map.

Proof. See [13], Lemma 8.1. Remark that Burns and Greither consider complexes over Λ⊗Zp Qp but sinceH is a Bézout ring, all their arguments work in our case and are omitted here.

5.2.6.3. Now we can prove Theorem 5. By Theorem 4 the complexRΓ

(η0)Iw,h (D,V ) is semisimple, its cohomology is H-torsion and the first assertion follows from Lemma 4

together with Corollary 1.

5.2.6.4. In this subsection we prove ii). Define

RΓ (Qv,N,T ) = RΓ(η0)

Iw (Qv,N,T )⊗LΛ Zp,

RΓ (Qv,D,V ) = RΓ (Qv,N,T )⊗Zp Qp.

We remark that for v = p this definition coincides with the definition given in Sect. 5.2.2. Applying ⊗LHQp

to the mapRΓ

(η0)Iw (Qv,N,T )⊗L

Λ H→ RΓ(η0)

Iw (Qv,T )⊗LΛ H

we obtain a morphismRΓf (Qv,D,V )→ RΓ (Qv,V ).

If v 6= p, then RΓf (Qv,D,V )=RΓf (Qv,V ) and this morphism coincides with the natural map RΓf (Qv,V )→RΓ (Qv,V ). If v = p, then RΓf (Qv,D,V ) = D[−1] and by Theorem 3 it coincides with the composition

D1−p−1ϕ−1

1−ϕ−−−−−−→ Dcris(V )(h−1)!expV,Qp−−−−−−−−→ H1(Qp,V ).


Since RΓ S(V ) = RΓ(η0)

Iw,S (V )⊗LΛQp this implies that

RΓ(η0)

Iw,h (D,V )⊗HQp = RΓh(D,V ) (39)

where RΓh(D,V ) is the Selmer complex associated to the diagram

RΓS(V ) // ⊕v∈S

RΓ (Qv,V )

⊕v∈S

RΓ (Qv,D,V ).

OO

We have a distinguished triangle

RΓh(D,V )→ RΓS(V )⊕(⊕

v∈S

RΓ (Qv,D,V )

)→

→⊕v∈S

RΓ (Qv,V )→ RΓh(D,V )[1]. (40)

Passing to determinants and using the trivialisation (10) of RΓf (Qv,V ) for v 6= p we obtain isomorphisms

det−1Qp

RΓS(V )⊗Qp

(⊗

v∈SdetQpRΓ (Qv,V )

)⊗detQpD ∼→ det−1

QpRΓh(D,V ),

ξD,h : ∆EP(V )⊗Qp

(detQpD⊗det−1

QpV+)∼→ det−1

QpRΓh(D,V ). (41)

where ∆EP(V ) is the Euler–Poincaré line defined in Sect. 4.1.1. From (39) it follows that for any i one hasan exact sequence

0→ RiΓ

(η0)Iw,h (D,V )Γ1 → Ri

Γh(D,V )→ Ri+1Γ

(η0)Iw,h (D,V )Γ1 → 0.

Now by Theorem 4

RiΓh(D,V ) =

R2Γ

(η0)Iw,h (D,V )Γ1 if i = 1

R2Γ(η0)

Iw,h (D,V )Γ1 if i = 20 if i 6= 1,2.

(42)

By Corollary 1 the projection R2Γ(η0)

Iw,h (D,V )Γ1 → R2Γ(η0)

Iw,h (D,V )Γ1 is an isomorphism which induces acanonical trivialisation

ϑD,h : det−1Qp

RΓh(D,V )∼→Qp.

of RΓh(D,V ). Applying Lemma 5.2.6.2 to the complex RΓ(η0)

Iw,h (D,V ) we obtain a commutative diagram

det−1H RΓ

(η0)Iw,h (D,V )

X−eiV,Iw,h //

⊗LHQp

H0

det−1

QpRΓh(D,V )

ϑD,h // Qp

(43)

where iV,Iw,h is the canonical trivialisation constructed in Sect. 5.2.1. From the definition of the module ofp-adic L-functions

94 Denis Benois

∆Iw,h(N,T )⊗LΛ Zp

∼→

det−1Zp

RΓS(T )⊗(⊗

v∈SdetZpRΓ (Qv,T )

)⊗

(⊗

v∈S−pdetZpRΓf (Qv,T )

)⊗detZpN

and again the canonical trivialisation (10) of RΓf (Qv,V ) for v 6= p induces a map

∆Iw,h(N,T )⊗LΛ Zp → det−1

QpRΓh(D,V ).

Comparing this map with the definition (41) of ξD,h one has

∆Iw,h(N,T )⊗LΛ Zp = ξD,h

(∆EP(T )⊗Zp ωN⊗Zp (ω

+T )−1) .

By definition, LIw(T,N,s) is a generator of iV,Iw,h(∆Iw,h(N,T )). Therefore, the diagram (43) gives

ϑD,h ξD,h(∆EP(T )⊗Zp ωN⊗Zp (ω+T )−1) = log(χ(γ))−eL∗Iw,h(T,N,0)Zp. (44)

We will now compute the left hand side of this equality in terms of the canonical trivialisation of the Euler–Poincaré line. Roughly speaking we should compare the trivialisation of the Euler–Poincaré line to thetrivialisation ϑD,h. Consider the commutative diagram

RΓf (V ) // RΓS(V )⊕ ⊕

v∈S∪∞RΓf (Qv,V ) // ⊕

v∈S∪∞RΓ (Qv,V )

RΓh(D,V ) //

OO

RΓS(V )⊕⊕

v∈SRΓ (Qv,D,V ) //

OO

⊕v∈S

RΓ (Qv,V )

OO

L //

OO

S⊕V+[−1] //

OO

V+[−1]

OO

(45)

where L = cone(RΓh(D,V )→ RΓf (V )

)[−1] and S is defined by (33). The upper and middle rows of (45)

coincide with (8) and (40) up to the following modification: the map locp : RΓf (Qp,V )→ RΓ (Qp,V ) isreplaced by Γ (h) locp. Hence S is isomorphic to L in the derived category Dp(Qp) and we have an exacttriangle

S→ RΓh(D,V )→ RΓf (V )→ S[1]. (46)

The cohomology of S is computed in (35). On the other hand, Corollary 1 together with (42) give

RiΓh(D,V )

∼→

Dϕ=1 if i = 1,

H1(Qp,V )

H1f ,p(V )+H1(D−1)

if i = 2.

An easy diagram chase shows that the map H1(S)→ R1Γh(D,V ) induced by (46) coincides with theidentity map id : Dϕ=p−1 → Dϕ=p−1

and that one has an exact sequence

0→ H1f (V )→ H2(S)→ R2

Γh(D,V )→ 0

which can be identified with

0→ H1f (V )→ Dcris(V )

Fil0Dcris(V )+D−1

Γ (h)expV−−−−−→H1(Qp,V )

H1f ,p(V )+H1(D−1)

→ 0.

Therefore, we have a commutative diagram


detQpS⊗detQpRΓf (V )ν //

βS

detQpRΓh(D,V )

ϑ∗D,h

Qp

κ // Qp

(47)

where βS was defined in (36), ϑ ∗D,h is the dual of the trivialisation ϑD,h and κ is the unique map which makesthis diagram commute.

From Proposition 7 and Corollary 1 ii) giving an explicit description of the trivialisation R1Γh(D,V )→R2Γh(D,V ) we obtain immediately that

κ = (log χ(γ))e(

1− 1p

)e

L (V,D)−1 idQp . (48)

Passing to determinants in the diagram (45) we obtain a commutative diagram

∆EP(V )⊗(

detQp tV (Qp)⊗det−1Qp

V+)⊗detQp RΓf (V )

αS

// Qp

=

∆EP(V )⊗ (detQp D⊗det−1Qp

V+)⊗detQp RΓh(D,V )⊗ (detQp S⊗det−1Qp

L)

ξD,h⊗µ

det−1

QpRΓh(D,V )⊗detQp RΓh(D,V )

duality // Qp

(49)

where αS is defined by (34) and µ : detS⊗det−1L→Qp is induced by (46). The upper map can be easilycompared with the trivialisation map iωM ,p defined by (21): it sends ∆EP(T )⊗ (ωtM ⊗ (ω+

T )−1⊗ω f ) onto

Γ (h)d+(V ) iωM ,p(∆EP(T ))

Ω(et,p)M (ω+

T ,ω+B )

. (50)

The tautological isomorphism detQpL' detQpRΓh(D,V )⊗det−1Qp

RΓf (V ) gives a commutative diagram

detQpRΓh(D,V )⊗detQp S⊗det−1Qp

L //

µ

detQpS⊗detQpRΓf (V )

ν

detQpRΓh(D,V )

= // detQpRΓh(D,V )

(51)

We can summarize the diagrams (49), (51) and (47) in the following commutative diagram

∆EP(V )⊗(det(tV (Qp))⊗det−1V+

)⊗detRΓf (V )

αS

// Qp

∆EP(V )⊗ (detD⊗detS⊗detRΓf (V ))⊗det−1V+ξD,h⊗ν

//

id⊗βS

det−1RΓf ,h(D,V )⊗detRΓf ,h(D,V )duality //

id⊗ϑ∗D,h

Qp

∆EP(V )⊗(

detD⊗det−1QpV+

) ξD,h⊗κ

// det−1RΓf ,h(D,V )ϑD,h // Qp.

The image of ∆EP(T )⊗ (ωtM ⊗ (ω+T )−1⊗ω f ) under the upper map is given by (50). The composition of

the left vertical maps is the map θS defined by (37). By Lemma 3 θS sends ∆EP(T )⊗ (ωtM ⊗ (ω+T )−1⊗ω f )

onto

det∗(

1− p−1ϕ−1

1−ϕ| D)−1

Ep(V,1)−1RV,D(ωV,N) ∆EP(T )⊗ (ωN⊗ (ω+T )−1). (52)

Next, (44) and (48) give

96 Denis Benois

ϑD,h (ξD,h⊗κ)(∆EP(T )⊗ωN⊗ (ω+T )−1) =

=

(1− 1

p

)e

L (V,D)−1L∗Iw,h(T,N,0)Zp. (53)

Putting together (50), (52) and (53) we obtain that

L∗Iw,h(T,N,0)

RV,D(ωV,N)∼p∼p Γ (h)d+(V )L (V,D) Ep(V,1)detQp

(1− p−1ϕ−1

1−ϕ|D−1

)iωM ,p (∆EP(T ))

Ω(et,p)M (ω+

T ,ω+B )

and the theorem is proved. ut

5.2.5 Special values of L∗Iw,h(T,N,s)

Let H1f (T ) denote the image of H1

f (T ) in H1f (V ) and let ωT, f be a base of

detZp H1f (T ). Let RV,D(ωT,N) denote the determinant of rV,D computed in the bases ωtM , ωN and ωT, f .

Corollary 2. Under the assumptions of Theorem 5 one has

L∗Iw,h(T,N,0)

RV,D(ωT,N)∼p Γ (h)d+(V )L (V,D)E+(V,D)

#III(T ∗(1))Tam0ωM

(T )

#H0S (V/T )#H0

S (V∗(1)/T ∗(1))

,

where III(T ∗(1)) is the Tate–Shafarevich group of Bloch–Kato [4] and Tam0ωM

(T ) is the product of localTamagawa numbers of T taken over all primes and computed with respect to a fixed base ωtM of detQtM(Q).

Proof. The computation of the trivialisation of the Euler–Poincaré line (see for example [31], Chap. II,Théorème 5.6.3) together with the definition (21) of iωM ,p give

iωM ,p(∆EP(T )) =#III(T ∗(1))Tam0

ωM(T )

#H0S (V/T )#H0

S (V∗(1)/T ∗(1))

Ω(et,p)M (ω+

T ,ω+MB

) [ω f : ωT, f ].

Since RV,D(ωT,N) = RV,D(ωV,N) [ω f : ωT, f ] the corollary follows fromTheorem 5.

5.2.6 The functional equation

Recall that we set hi(V ) = dimQp(griDdR(V )) and m = ∑i∈Z

ihi(V ). Since V is crystalline, detQp(V ) is a one

dimensional crystalline representation and detZp(T ) = T0(m) where T0 is an unramified GQp -module of rank1 over Zp. The module (T0⊗W (Fp))

ϕ=1 em where em = (t−1⊗ ε)⊗m is a Zp-lattice in detQp(Dcris(V )) =Dcris(V0(m)) which depends only on T and which we denote by Dcris(T0(m)).

Let D⊥ be the dual regular module. The exact sequence

0→ D→ Dcris(V )→(

D⊥)∗→ 0

gives an isomorphismdetQpD⊗det−1

QpD⊥ ' detQpDcris(V )

and we fix a lattice N⊥ ⊂ D⊥ such that

detZpN⊗det−1Zp

N⊥ ' Dcris(T0(m)).

Set ΓV,h(s) = ∏j>−h

( j + s)dimFil jDdR(V ). The conjecture δZp(V ) of [25] proved in [BB12] implies that for

h 0


LIw,h(T ∗(1),N⊥,−s)∼Λ∗ Γ−1

V,h (s) ∏−h< j<h

( j+ s)d+(V ∗(1))LIw,h(T,N,s)

(see [55], Théorème 2.5.2). This can seen as the algebraic counterpart of the functional equation for p-adicL-functions. An elementary computation (see [BB12], Lemme 4.7) shows that

Γ−1

V,h (s) ∏−h< j<h

( j+ s)d+(V ∗(1)) = Γ∗(V )Γ (h)d+(V ∗(1))−d+(V )sr +o(sr).

where r = dimQp tV (Qp)−d+(V )= dimQp H1f (V ) and Γ ∗(V ) is defined by (23). Therefore LIw,h(T ∗(1),N⊥,s)

has a zero of order dimQp H1f (V )+ e at s = 0. Moreover one has

L∗Iw,h(T∗(1),N⊥,0)

Γ (h)d+(V ∗(1))∼p Γ

∗(V )LIw,h(T,N,0)

Γ (h)d+(V ).

From the definition of RV ∗(1),D⊥ (see Sect. 4.2.1) one has

RV ∗(1),D⊥(ωV ∗(1),N⊥) = Ω(H,p)M (ωT ,ωMdR)

−1RV,D(ωV,N)

where Ω(H,p)M denotes the period map defined by (17) and (18) and ωMdR =ωtM⊗ω

−1tM∗(1) . Taking into account

(24) we obtain that

L∗Iw,h(T∗(1),N⊥,0)

RV ∗(1),D⊥(ωV,N⊥)∼p Γ (h)d+(V ∗(1))L (V,D)E+(V ∗(1),D⊥)

iωM ,p (∆EP(T ∗(1)))

Ω(et,p)M (ω+

T ∗(1),ω+M∗(1)B

),

which is the analogue of Theorem 5 for LIw,h(T ∗(1),N⊥,s).

Appendix

In this Appendix we prove some results about the cohomology of p-adic representations of local fields.Let K be a finite extension of Qp and T a p-adic representation of GK . Fix a topological generator γ of

Γ . Let D(T ) = (T ⊗Zp A)HK be the (ϕ,Γ )-module associated to T by Fontaine’s theory [27]. Consider thecomplex

Cϕ,γ(D(T )) =[D(T )

f−→ D(T )⊕D(T )g−→ D(T )

]where the modules are placed in degrees 0, 1 and 2 and the maps f and g are given by

f (x) = ((ϕ−1)x , (γ−1)x), g(y,z) = (γ−1)y − (ϕ−1)z.

Proposition 8. There are canonical and functorial isomorphisms

hi : H i(Cϕ,γ(D(T ))) ∼→ H i(K,T )

which can be described explicitly by the following formulas:i) If i = 0, then h0 coincides with the natural isomorphism

D(T )ϕ=1,γ=1 = H0(K,T ⊗Zp Aϕ=1) = H0(K,T ).

ii) Let α,β ∈D(T ) be such that (γ−1)α = (1−ϕ)β . Then h1 sends cl(α,β ) to the class of the cocycle

µ1(g) = (g−1)x +g−1γ−1

β ,

where x ∈ D(T )⊗AK A is a solution of the equation (1−ϕ)x = α.

98 Denis Benois

iii) Let γ ∈GK be a lifting of g ∈Γ and let x be a solution of (ϕ−1)x = α. Then h2 sends α to the classof the 2-cocycle

µ2(g1,g2) = γk1(h1−1)

γk2 −1γ−1

x

where gi = γkihi, hi ∈ HK .

Proof. The isomorphisms hi were constructed in [40], Theorem 2.1. Remark that i) follows directly fromthis construction (see [40], p.573) and that ii) is proved in [2], Proposition 1.3.2 and [15], Proposition I.4.1.The proof of iii) follows along exactly the same lines. Namely, it is enough to prove this formula modulopn for each n. Let α ∈ D(T )/pnD(T ). By Proposition 2.4 of [40] there exists r ≥ 0 and y ∈ D(T )/pnD(T )such that (ϕ−1)α = (γ−1)rβ . Let

Nx = (D(T )/pnD(T ))⊕ (⊕ri=1(AK/pnAK) ti),

where ϕ(ti) = ti+(γ−1)r−i(α) and γ(ti) = ti+ ti−1. Then Nx is a (ϕ,Γ )-module and we have a short exactsequence

0→ D→ Nx→ X → 0

where X = Nx/M ' ⊕ri=1AK/pnAK ti. An easy diagram chase shows that the connecting homomorphism

δ 1D : H1(Cϕ,γ(D(X)))→ H2(Cϕ,γ(D(T ))) sends cl(0, tr) to −cl(α). The functor V(D) = (D⊗AK A)ϕ=1 is

a quasi-inverse to D. Thus one has an exact sequence of Galois modules

0→ T/pnT → Tx→ V(X)→ 0

where Tx =V(Nx). From the definition of x it follows immediately that tr−x∈ Tx. By ii), h1(cl(0, tr)) can be

represented by the cocycle c(g) =g−1γ−1

tr and we fix its lifting c : GK → Nx putting c(g) =g−1γ−1

(tr− x).

As g1c(g2)− c(g1g2) + c(g1) = −µ2(g1,g2), the connecting map δ 1T : H1(K,V(X)) → H2(K,T/pnT )

sends cl(c) to −cl(µ2) and iii) follows from the commutativity of the diagram

H1(Cϕ,γ(X))δ 1

D //

h1

H2(Cϕ,γ(T/pnT ))

h2

H1(K,V(X))

δ 1T // H2(K,T/pnT ).

Proposition 9. The complexes RΓ (K,T ) and Cϕ,γ(T ) are isomorphic in D(Zp).

Proof. The proof is standard (see for example [12]). The exact sequence

0→ T → D(T )⊗AK A ϕ−1−−→ D(T )⊗AK A→ 0

gives rise to an exact sequence of complexes

0→C•c (GK ,T )→C•c (GK ,D(T )⊗AK A)ϕ−1−−→C•c (GK ,D(T )⊗AK A)→ 0

Thus RΓ (K,T ) is quasi-isomorphic to the total complex

K•(T ) = Tot•(

C•c (GK ,D(T )⊗AK A)ϕ−1−−→C•c (GK ,D(T )⊗AK A)

).

On the other handCϕ,γ(T ) = Tot•

(A•(T )

ϕ−1−−→ A•(T )),

where A•(T ) = [D(T )γ−1−−→ D(T )]. Consider the following commutative diagram of complexes


D(T )

β0

γ−1 // D(T )

β1

// 0 //

· · ·

C0(GK ,D(T )⊗AK A) // C1(GK ,D(T )⊗AK A) // C2(GK ,D(T )⊗AK A) // · · ·

in which β0(x) = x viewed as a constant function on GK and β1(x) denotes the map GK → D(T )⊗AK A

defined by (β1(x))(g) =g−1γ−1

x. This diagram induces a map Tot•(A•(T )ϕ−1−−→ A•(T ))→ K•(T ) and we

obtain a diagramCϕ,γ(T )→ K•(T )← RΓ (K,T )

where the right map is a quasi-isomorphism. Then for each i one has a map

H i(Cϕ,γ(T ))→ H i(K•(T ))' H i(K,T )

and an easy diagram chase shows that it coincides with hi. The proposition is proved.

Corollary 3. Let V be a p-adic representation of GK . Then the complexes RΓ (K,V ), Cϕ,γ(D†(V )) andCϕ,γ(D†

rig(V )) are isomorphic in D(Qp).

Proof. This follows from Theorem 1.1 of [18] together with Proposition A.2.

Recall that K∞/K denotes the cyclotomic extension obtained by adjoining all pn-th roots of unity. LetΓ = Gal(K∞/K) and let Λ(Γ ) = Zp[[Γ ]] denote the Iwasawa algebra of Γ . For any Zp-adic representationT of GK the induced representation IndK∞/KT is isomorphic to (T ⊗Zp Λ(Γ ))ι and we set RΓIw(K,T ) =C•c (GK , IndK∞/KT ). Consider the complex

CIw,ψ(T ) =[D(T )

ψ−1−−→ D(T )]

in which the first term is placed in degree 1.

Proposition 10. There are canonical and functorial isomorphisms

hiIw : H i(CIw,ψ(T ))→ H i

Iw(K,T )

which can be described explicitly by the following formulas:i) Let α ∈D(T )ψ=1. Then (ϕ−1)α ∈D(T )ψ=0 and for any n there exists a unique βn ∈D(T ) such that

(γn−1)βn = (ϕ−1)α. The map h1Iw sends cl(α) to (h1

n(cl(βn,α)))n∈N ∈ H1Iw(Kn,T ).

ii) If α ∈ D(T ), then h2Iw(cl(α)) = −(h2

n(ϕ(α)))n∈N.

Proof. The proposition follows from Theorem II.1.3 and Remark II.3.2 of [15] together with PropositionA.1.

Proposition 11. The complexes RΓIw(K,T ) and CIw,ψ(T ) are isomorphic in the derived category D(Λ(Γ )).

Proof. We repeat the arguments used in the proof of Proposition A.2 with some modifications. For anyn> 1 one has an exact sequence

0→ IndKn/KT → (D(T )⊗Zp Zp[Gn]ι)⊗AK A ϕ−1−−→ (D(T )⊗Zp Zp[Gn]

ι)⊗AK A→ 0.

Set D(IndK∞/KT ) = D(T )⊗Zp Λ(Γ )ι and

D(IndK∞/K(T ))⊗AK A = lim←−n(D(T )⊗Zp Zp[Gn]

ι)⊗AK A.

As IndKn/KT are compact, taking projective limit one obtains an exact sequence

0→ IndK∞/KT → D(IndK∞/K(T ))⊗AK A ϕ−1−−→ D(IndK∞/K(T ))⊗AK A→ 0.

100 Denis Benois

Thus RIw(K,T ) is quasi-isomorphic to

K•Iw(T ) = Tot•(

C•c (GK ,D(IndK∞/KT )⊗AK A)ϕ−1−−→C•c (GK ,D(IndK∞/KT )⊗AK A)

).

We construct a quasi-isomorphism f• : CIw,ψ(T )→ K•Iw(T ). Any x ∈ D(T ) can be written in the formx= (1−ϕψ)x + ϕψ(x) where ψ(1−ϕψ)x = 0. Then for each n> 0 the equation (γn−1)yn = (ϕψ−1)x

has a unique solution yn ∈ D(T )ψ=0 ([15], Proposition I.5.1). In particular, yn =γn+1−1γn−1

yn+1 and we

have a compatible system of elements Yn =|Gn|−1

∑k=0

γk ⊗ γ

k(yn) ∈ D(T )⊗Zp Zp[Gn]ι . Put Y = (Yn)n≥0 ∈

D(IndK∞/KT ). Then(γn−1)Yn = (γ−1)Y (mod D(IndKn/KT )).

Let ηx ∈C1c (GK ,D(IndK∞/KT )⊗AK A) be the map defined by ηx(g) =

g−1γ−1

(1⊗ x). Define

f1 : D(T )→ K1Iw(T ) = C0

c (GK ,D(IndK∞/KT )⊗AK A)⊕C1c (GK ,D(IndK∞/KT )⊗AK A)

by f1(x) = (Y,ηx) and

f2 : D(T )→C1c (GK ,D(IndK∞/KT )⊗AK A)⊂ K2

Iw(T )

by f2(z) =−ηϕ(z). It is easy to check that f• is a morphism of complexes. This gives a diagram

CIw,ψ(T )→ K•Iw(T )← RΓIw(K,T )

in which the right map is a quasi-isomorphism. Using Proposition A.4 it is not difficult to check that foreach i the induced map

H i(CIw,ψ(T ))→ H i(K•Iw(T ))' H iIw(K,T )

coincides with hiIw. The proposition is proved.

Corollary 4. The complexes RΓIw(K,T ) and C†Iw,ψ(T ) are isomorphic in

D(Λ(Γ )).

Proof. One has D†(T )ψ=1 =D(T )ψ=1 ([14], Proposition 3.3.2) and D†(T )/(ψ−1) =D(T )/(ψ−1) ([18],Lemma 3.6). This shows that the inclusion C†

Iw,ψ(T )→CIw,ψ(T ) is a quasi-isomorphism.

remark These results can be slightly improved. Namely, set rn = (p−1)pn−1. The method used in the proofof Proposition III.2.1 [15] allows to show that ψ(D†,rn(T )) ⊂ D†,rn−1(T ) for n 0. Moreover, for anya ∈ D†,rn(T ) the solutions of the equation (ψ−1)x = a are in D†,rn(T ). Thus

C†,rnIw,ψ(T ) =

[D†,rn(T )

ψ−1−−→ D†,rn(T )], n 0

is a well-defined complex which is quasi-isomorphic to C†Iw,ψ(T ). Further, as ϕ(A†,r/p) = A†,r we can

consider the complex

C†,rnϕ,γ (T ) =

[D†,rn−1(T )

f−→ D†,rn(T )⊕D†,rn−1(T )g−→ D†,rn(T )

], n 0

in which f and g are defined by the same formulas as before. Then the inclusion C†,rnϕ,γ (T )→Cϕ,γ(T ) is a

quasi-isomorphism.

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archiv.ub.uni-heidelberg.de/volltextserver/14935/1/diss?




On Special L-Values attached to Siegel Modular Forms.

Thanasis Bouganis

Abstract In his admirable book “Arithmeticity in the Theory of Automorphic Forms” Shimura establishesvarious algebraicity results concerning special values of Siegel modular forms. These results are all statedover an algebraic closure of Q. In this article we work out the field of definition of these special values. Inthis way we extend some previous results obtained by Sturm, Harris, Panchishkin, and Böcherer-Schmidt.

1 Introduction

Special values of L-functions play a central role in Iwasawa theory since they are indispensable for theformulation of the Main Conjectures. It is precisely this information which is encoded in the interpolationproperties of the p-adic L-functions. The first step to construct these p-adic L-functions is to show that theL-functions under consideration evaluated at “critical” points have particular algebraic properties. Theseproperties are usually described by Deligne’s conjectures. In this article we address this kind of questionsfor L-functions associated to Siegel modular forms.

This article grew out of the author’s effort to read carefully the book of Shimura “Arithmeticity in theTheory of Automorphic Forms” ([23]) which means to do also the “exercises” left by Shimura to the reader.One of them is related to the algebraicity of various special values of Siegel modular forms (see page 239,Remark 28.13 in (loc. cit.)). As Shimura points out the results left as exercises should follow by using thevarious techniques and results obtained in his book and various papers of him. This is indeed the case sincemost ideas of this article can be found in the various works of Shimura, which of course in turn requiressome familiarity with them. In any rate we believe that it is useful to have the results worked out in this paperdocumented in the literature, and for this reason we decided to write this article. In this paper we considerthe special values of Siegel modular forms of integral weight. In [5], the continuation of this article, weconsider also special L-value attached to half-integral weight Siegel modular forms.

Let us point out some results in this article that we believe deserve special mention. The first is thereciprocity law of the action of the Galois group on half-integral weight Eisenstein series. For integralweight Eisenstein series one can find the reciprocity laws in the book of Feit ([8]) (if not in the form that itis needed for our purposes). However to the best of our knowledge the reciprocity for half integral Eisensteinseries has not been worked out for Siegel modular forms. Another interesting result is the definition of theperiod Ωf appearing in Theorem 12. These kind of periods have been first considered by Sturm and Harris[25, 9] (and later also by Panchishkin), based on an idea of Shimura. We follow the ideas of Sturm indefining them but using some new results of Shimura we are able to improve in some cases the bounds onthe weight of the Siegel modular forms that the results are applicable. Also the fact that we use the moreprecise form of the Andrianov-Kalinin type identity proved by Shimura, we can obtain slightly finer results,since we need to remove less Euler factors of the L-function.

This paper is organized as follows. In section two we have a very brief introduction to Siegel modularforms. Then we move to section three where after presenting various results of Shimura with respect the

Thanasis BouganisDepartment of Mathematical Sciences, Durham University, South Road, DH1 3LE Durham, U.K.e-mail: [email protected]

103

[email protected]

[email protected]

104 Thanasis Bouganis

theory of theta series and Eisenstein series for the symplectic group, we prove the various reciprocity lawsof the action of the absolute Galois group on the Eisenstein series. Some of the result have already appearedin [25] and [8], and we use ideas of these works. For the case of half integral weight Eisenstein series weprove the reciprocity inspired by an idea of Shimura. In section 4 we introduce the L-functions which areconsidered in this paper. All the material of this section is from Shimura’s book. In section 5 we also presentthe work of Shimura on the generalization of the so-called Adrianov-Kalinin type identity. However for ourpurposes we use an integral expression that it is not in the book [23] but in a paper of Shimura [20]. Theuse of this integral expression will lead to study slightly different L-functions than in the ones studied in thebook of Shimura (we explain more later on this). Also in this section all the material is taken from works ofShimura. In section 6 we define the periods that we will use to obtain the good reciprocity laws. The idea ofdefining this periods as values of an L-function goes back to Shimura, and have been used by Sturm [25],Harris [9] and Panchishkin [17] in the case Siegel modular forms over the rationals and of even degree.We also note that we obtain a slightly different results than in these works, partly because we use somenewer results of Shimura that were not available when these works were written. In the following sectionwe present the various results on the field of rationality of the various special functions properly normalizedand in some cases we provide some reciprocity results. Finally we finish this work by briefly discussing yetanother method for considering the same questions as in this paper, namely the doubling method.

One last remark with respect to the notation used in this article. Since we are using as our main referencethe book of Shimura [23] we decided to keep, the anyway excellent, notation used by him. In particular ifsome times we use some notions not defined in this paper the reader will find the exact same notation alsoin the reference. This allows to keep the length of this article reasonable since we do not need to introduceall the mathematical notions used here. We finally remark that our choice to use the notation as in Shimura’sbook leads us to write Z(s, f) for the L-functions attached to a Siegel modular form f instead of the morestandard L(s, f).

2 Siegel Modular forms

2.1 Integral weight Siegel modular forms

In this section we introduce the notion of a Siegel modular form (classicaly and adelically). We followclosely the book of Shimura [23].

For a positive integer n ∈ N we define the matrix ηn :=(

0 −1n1n 0

)and for any commutative ring A

with an identity the group Spn(A) := α ∈ GL2n(A)|tαηnα = ηn. The group Spn(R) acts on the Siegelupper half space Hn := z ∈ Cn

n|tz = z, Im(z) > 0 by linear fractional transformations, that is for α =(aα bα

cα dα

)∈ Spn(R) and z ∈ Hn we have α · z := (aα z+ bα)(cα z+ dα)

−1 ∈ Hn. Moreover if we define

µα(z) := µ(α,z) := cα z+dα then we have

µ(βα,z) = µ(β ,αz)µ(α,z), α,β ∈ Spn(R),z ∈Hn

Let now F be a totally real field of degree d := [F : Q] and write g for its ring of integers. We write afor the set of archimedean places of F , h for the finite ones and we set G := Spn(F). We write GA for theadelic group and we decompose GA = GhGa where Ga := ∏v∈a Gv and Gh := ∏v∈h Gv. For two fractionalideals a and b of F such that ab⊆ g, we define the subgroup of GA,

D[a,b] :=

x =(

ax bxcx dx

)∈ GA|ax ≺ gv,bx ≺ av,cx ≺ bv,dx ≺ gv,∀v ∈ h

,

where we use the notation “≺” of Shimura, x ≺ bv meaning that the v-component of x is a matrix withentries in the ideal bv. We will mainly consider groups of the form D[b−1,bc] for a fractional ideal b andan integral ideal c. Strong approximation for G implies that GA = GqD[b−1,bc] for any b,c and q ∈Gh. Wedefine Γ q(b,c) := G∩qD[b−1,bc]q−1. Given a Hecke character ψ of F with ψv(a) = 1 for all a ∈ g×v , v ∈ h

On Special L-Values attached to Siegel Modular Forms. 105

such that a−1 ∈ cv we define a character on D[b−1,bc] by ψ(x) = ∏v|c ψv(det(dx)v) and a character whichwe still denote ψ on Γ q by ψ(γ) := ψ(q−1γq).

We now write Za := ∏v∈aZ and H := ∏v∈aHn. For a function f : H→ C and an element k ∈ Za wedefine

( f |kα)(z) := jα(z)−k f (αz), α ∈ G, z ∈H.

Here we write z = (zv)v∈a with zv ∈ Hn and αv ∈ Spn(R) and define jα(z)−k := ∏v det(µαv(zv))−kv . Let

now Γ be group of the form Γ q, q ∈ Gh as above and ψ a Hecke character. Then we define

Definition 1. A function f : H→ C is called a Siegel modular form for the congruence subgroup Γ ofweight k ∈ Za and Nebentypus ψ if

1. f is holomorphic,2. f |kγ = ψ(γ) f for all γ ∈ Γ ,3. f is holomorphic at cusps.

The last condition is needed only if F = Q and n = 1. Then it is the classical condition of elliptic mod-ular forms being holomorphic at cusps. The above defined space we will denote it by Mk(Γ ,ψ). As it isexplained in [23, page 33] for an element f ∈Mk(Γ ,ψ) and an element α ∈G we have a Fourier expansion

( f |kα)(z) = ∑h∈S+

cα(h)ena(hz),

where S+ is the set of n by n symmetric matrices with entries in F which are positive semi-definite atevery real place v ∈ a and en

a(x) = exp(2πi∑v∈a tr(xv)). An element f ∈Mk(Γ ,ψ) is called a cusp form ifcα(h) 6= 0 for some α ∈ G implies hv is positive definite for all v ∈ a.

We now turn to the adelic Siegel modular forms. Let D be a group of the form D[b−1,bc] and ψ a Heckecharacter of F .

Definition 2. A function f : GA→ C is called adelic Siegel modular form if

1. f(αxw) = ψ(w) jkw(i)f(x) for α ∈ G, w ∈ D with w(i) = i,

2. For every p ∈ Gh there exists fp ∈Mk(Γp,ψp), where Γ p := G∩ pDp−1 and ψp(γ) = ψ(pγ p−1) such

that f(py) = ( fp|ky)(i) for every y ∈ Ga.

We write Mk(D,ψ) for this space. Strong approximation theorem for Spn gives Mk(D,ψ)∼= Mk(Γq,ψq) for

any q ∈Gh. We define the space of automorphic cusp form Sk(D,ψ) to be the subspace of Mk(D,ψ) that isin bijection with Sk(Γ

q,ψ) for any q ∈Gh in the above bijection. We may also sometimes write Mk(b,c,ψ)for Mk(D,ψ). Similarly we may write Mk(b,c,ψ) for Mk(Γ ,ψ) where Γ = G∩D[b−1,bc], i.e q = 1.

2.2 Half-integral weight Siegel modular forms

Even though we will consider only algebraicity results for integral weight Siegel modular forms, in manycase we will need to use half-integral weight modular forms. We denote by MA the adelized metaplecticgroup sitting in the exact sequence 0→ T→ MA → GA → 0. The last projection we denote by pr. Wewrite Cθ for the theta group defined for example in [20, page 536] and Γ θ = G∩Cθ . We also define thegroup M = x ∈MA|pr(x) ∈ PACθ, where P is the standard Siegel parabolic subgroup of G. Thanks to acanonical lift we may consider G as a subgroup of MA and hence also Γ θ a subgroup of M. For an elementσ ∈M and z ∈ H we write hσ (z) for the holomorphic function defined by Shimura. By a half integralweight k ∈ 1

2Za we mean a tuple (kv)v∈a so that kv ∈ Z+ 1

2 for all v ∈ a. For such a k we define the factor ofautomorphy

jσ (z)k := hσ (z) jpr(σ)(z)[k].

Then the definition of half integral weight modular forms, with congruence subgroup Γ ≤ Γ θ is the sameas in integral case but using the new factor of automorphy. One may define also adelic automorphic forms,we refer to Shimura [23, page 166] for this.


3 Theta and Eisenstein Series

3.1 Theta series

Following Shimura (see page 270 in [23]) we set W = Fnn and we let S(Wh) denote the space of Schwartz-

Bruhat functions on Wh. Let τ be an n by n symmetric matrix with entries in F such that τv > 0 for all v ∈ a.For an element λ ∈ S(Wh) and an element µ ∈ Za such that 0≤ µv ≤ 1 for all v ∈ a we define

θ(z,λ ) = ∑ξ∈W

λ (ξh)det(ξ )µ ea(tr(tξ τξ z)), z ∈H.

It is shown in the appendix of [23] that this is an element of Ml with l := µ + n2 a. Moreover it is also shown

that if µ 6= 0 then θ(z,λ ) is actually a cusp form. We now introduce some extra notation following [23,Appendix A3.18]. We set

R = ∏vh(gv)

nn, Ev = GLn(gv), R∗ = RWa ⊂WA.

We let ω be now a Hecke character of F of conductor f such that ωa(−1)n = (−1)n∑v µv . Let now r be anelement of GLn(F)h and define

θ(z) := ∑W∩rR∗

ωa(det(ξ ))ω∗(det(r−1ξ )g)det(ξ )µ en

a(tξ τξ t),

where for a Hecke character ψ we denote by ψ∗ the corresponding ideal character. Then Shimura provesthe following proposition

Proposition 1 (Shimura). Let ρτ be the Hecke character of F corresponding to the extension F(c1/2)/Fwith c = (−1)[n/2]det(2τ); put ω ′ = ωρτ . Then there exist a fractional ideal b and an integral ideal c, suchthat the conductor of ω ′ divides c, D[b−1,bc]⊂ D[2d−1,2d] if n is odd, and

θ(γz) = ω′c(det(aγ)) jl

γ(z)θ(z), γ ∈ G∩D,

where D = x ∈ D[b−1,bc]. Moreover, if β ∈ G∩diag[q, q]C with q ∈ GLn(F)h, then

jlβ(β−1z)θ(β−1z) = ω

′(det(q))−1ω′c(det(dβ q))|det(q)|n/2

A ×

∑ξ∈W∩rR∗q−1

ωa(det(ξ ))ω∗(det(ξ r−1q)g)det(ξ )µ ena(

tξ τξ z).

In particular, let x and t be fractional ideals of F such that tg2τg ∈ x for every g ∈ rgn1 and th(2τ)−1h ∈ 4t−1

for every hrgn1 and write h for the conductor of ρτ . Then we can take

(b,c) =

(2−1dx,h∩ f∩ x−1f2t), if n is even;(2−1da−1,h∩ f∩4a∩af2t), if n is odd.

,

where a= x−1∩g.

3.2 Eisenstein series

We follow Shimura [23, pages 131-132] and define various Eisenstein series of Siegel type. Let k ∈ 12Z

a bea weight, b a fractional ideal of F , c an integral ideal in F and a Hecke character χ of F with infinity typeχa(x) = x`a|xa|−`, and

χv(a) = 1, if v ∈ h, a ∈ r×v , and a−1 ∈ rvcv, ∀ v ∈ h.


When k is half integral we also assume that D[b−1,bc] ⊂ D[2d−1,2d], where d the different ideal of F .Following the notation of Shimura we now define in the case of k ∈ Za

D = D[b−1,bc],

and otherwiseD = x ∈MA|pr(x) ∈ D[b−1,bc].

Write P = x ∈ G|cx = 0 for the standard Siegel parabolic. We then define a function µ on GA or MA by

µ(x) = 0, if x 6∈ PAD,

µ(x) = χh(det(dp))−1

χc(det(dw))−1 jk

x(i)−1| jx(i)|k,

if x = pw with p ∈ PA and w ∈ D. Then for a pair (x,s) ∈ GA×C if k ∈ Za or in MA×C otherwise, wedefine the Eisenstein series (for the function ε below we refer to Shimura’s book)

EA(x,s) = EA(x,s; χ, D) = ∑α∈P\G

µ(αx)ε(αx)−s.

We will need one more type of Eisenstein series. We define the element ζ ∈ Sp(n,F)A by

ζa = 1, ζh =

(0 −δ−11n

δ1n 0

),

where δ ∈ F×h such that δg = d. We further fix an element ζ ∈ MA such that pr(ζ ) = ζ and h(ζ ,z) = 1.Then we define the Eisenstein series

E∗A(x,s) = χ(δ )−n×

EA(xζ ,s), k ∈ Za ;EA(xζ ,s), otherwise.

Finally we define the Eisenstein series

DA(x,s) = E∗A(x,s)×

Lc(2s,χ)∏

[n/2]i=1 Lc(4s−2i,χ2), k ∈ Za;

∏[(n+1)/2]i=1 Lc(4s−2i−1,χ2), k 6∈ Za;

,

Write S = x ∈ Fnn |tx = x. Then the q-expansion of E∗A(x,s) is given by

E∗A

((q σ q0 q

),s)= ∑

h∈Sc(h,q,s)en

A(hσ),

where q ∈ GLn(F)A and σ ∈ SA. We now define the Eisenstein series on (z,s) ∈H×C by

E∗(x(i),s) = jkx(i)E

∗A(x,s),

and similarly we define D(z,s and D∗(z,s). When we want to indicate the dependence on the various inputdata we will write E(z,s;k,χ,c) for E(z,s) or in case we want also to indicate the dependency on b we willwrite E(z,s;k,χ,Γ ), where Γ = G∩D[b−1,bc]. We now note the q-expansion

E∗(z,s) = ∑h∈S

det(y)−ka/2c(h,q,s)ena(hx),

where qh = q1 and qa = y1/2. For the coefficients c(h,q,s) we have the following propositions of Shimura[23, Proposition 16.9, 16.10, 17.6],( for notation not introduced here we refer to (loc. cit)).

Proposition 2 (Shimura). Suppose that c 6= g and det(qv) > 0 for every v ∈ a. Then c(h,q,s) 6= 0 only if(tqhq)v ∈ (db−1c−1)vSv for every v ∈ h. In this case

c(h,q,s) =Cχh(det(−q))−1|det(q)h|n+1−2sA |DF |−2ns+3n(n+1)/4N(bc)−n(n+1)/2×


det(y)saΞ(y;h;sa+ k/2,sa− k/2)αe

c(ε−1b ·

tqhq,2s,χ),

where C = 1 and e = 0 if k ∈Za, and C = e(n[F : Q]/8) and e = 1 if k 6∈Za; εb ∈ F×h such that εbg= b−1d ifk ∈Za, and εb = 1 otherwise; DF is the discriminant of F.The function Ξ(g;h;α,β ) =∏v∈a ξ (yv,hv;αv,βv)is given in [23, page 140].

Proposition 3 (Shimura). Consider q and h such that c(h,q,s) 6= 0. Set r = rank(h) and let g ∈ GLn(F)such that g−1hg = diag[h′,0] with h′ ∈ Sr. Let ρh be the Hecke character corresponding to F(c1/2)/F wherec = (−1)[r/2]det(2h′), if r > 0; let ρh = 1 if r = 0. Then

αec(ε−1b ·

tqhq,2s,χ) = Λc(s)−1Λh(s)∏

v∈cfh,q,v

(χ(πv)|πv|2s+e/2

),

where

Λc(s) =

Lc(2s,χ)∏

[n/2]i=1 Lc(4s−2i,χ2), if k ∈ Za;

∏[(n+1)/2]i=1 Lc(4s−2i+1,χ2), otherwise.

Λh(s) =

Lc(2s−n+ r/2,χρh)∏

[(n−r)/2]i=1 Lc(4s−2n+ r+2i−1,χ2), if k ∈ Za;

∏[(n−r+1)/2]i=1 Lc(4s−2n+ r+2i−2,χ2), otherwise.

Here fh,q,v are polynomials with coefficients in Z, independent of χ . For the finite set c see [23].

For a number field W we follow Shimura and write Nrk(W ) for the space of W -rational nearly holomor-

phic forms of weight k (see [23, page 103 and page 110] for the definition). The theorem below is due toShimura [23, Theorem 17.9].

Theorem 1 (Shimura). Let Φ be the Galois closure of F over Q and let k ∈ 12Z

a with kv ≥ (n+1)/2 for allv∈ a and kv−kv′ ∈ 2Z for every v,v′ ∈ a. Let µ ∈ 1

2 Z with n+1−kv ≤ µ ≤ kv and |µ− n+12 |+

n+12 −kv ∈ 2Z

for all v ∈ a. Exclude the cases

1. µ = (n+2)/2, F =Q and χ2 = 1,2. µ = 0, c= g and χ = 1,3. 0 < µ ≤ n/2, c= g and χ2 = 1.

Then D(z,µ/2;k,χ,c) belongs to πβNrk(ΦQab), where r = (n/2)(k−|µ− (n+1)/2|a− n+1

2 a) except inthe case where n = 1, µ = 2, F =Q, χ = 1 and n > 1, µ = (n+3)/2, F =Q, χ2 = 1. In these two case wehave r = n(k−µ +2)/2. Moreover we have that β = (n/2)∑v∈a(kv +µ)− [F : Q]e where

e =[(n+1)2/4]−µ, if 2µ +n ∈ 2Z and µ ≥ λ ;[n2/4

], otherwise.

For an element p ∈ Za and a weight q ∈ 12Z

a we write ∆pq for the differential operators defined by

Shimura in [23, page 146]. In particular we have ∆pq N

tq(ΦQab) ⊂ πn|p|Nt+np

q+2p(ΦQab). Moreover for anyf ∈Nt

q(ΦQab) and any σ ∈ Gal(ΦQab/Φ) we have that(π−n|p|

∆pq ( f )

)σ

= π−n|p|

∆pq ( f σ ) (1)

Let µ ∈ 12Z and k ∈ 1

2Za be as in the theorem above. If µ ≥ (n+1)/2 then Shimura shows that [23, page

146]∆

pµaD(z,µ/2; µa,χ,c) = cp

µa(µ/2)(i/2)n|p|D(z,µ/2;ka,χ,c), (2)

where p = (k−µa)/2. Here cpµa(µ/2) ∈Q×. If µ < (n+1)/2 then we have

∆pνaD(z,µ/2;νa,χ,c) = cp

νa(µ/2)(i/2)n|p|D(z,µ/2;ka,χ,c), (3)

where ν = n+1−µ , p = (k−νa)/2 and again cpνa(µ/2) ∈Q×.

The following lemma is immediate from the above equations,

Lemma 1. Assume there exists A(χ),B(χ) ∈Qab and β1,β2 ∈ N such that for all σ ∈ Gal(Qab/Q)


(D(z,µ/2; µa,χ,c)

πβ1A(χ)

)σ

=D(z,µ/2; µa,χσ ,c)

πβ1A(χσ ), µ ≥ (n+1)/2

and (D(z,µ/2;νa,χ,c)

πβ2B(χ)

)σ

=D(z,µ/2;νa,χσ ,c)

πβ2B(χσ ), µ ≤ (n+1)/2.

Then we have for µ ≥ (n+1)/2 that(D(z,µ/2;k,χ,c)πβ1+n|p|in|p|A(χ)

)σ

=D(z,µ/2;k,χσ ,c)

πβ1+n|p|in|p|A(χσ ), p = (k−µa)/2 ∈ Za,

and for µ ≤ (n+1)/2 that(D(z,µ/2;k,χ,c)πβ2+n|p|in|p|B(χ)

)σ

=D(z,µ/2;k,χσ ,c)

πβ2+n|p|in|p|B(χσ ), ν = n+1−µ p = (k−νa)/2 ∈ Za,

We will be interested in algebraicity statements of the Eisenstein series of weight sufficient large it isenough to study the effect of the action of the Galois group of the full rank coefficients. More precisely wehave the following lemma.

Lemma 2. Let f (z) = ∑h∈S c(h)ena(hz) ∈ Mka(Qab) with k ≥ n/2. Assume that for an element σ ∈

Gal(Qab/Q) we have c(h)σ = ac(h) for all h with det(h) 6= 0 for some a ∈ C. Then c(h)σ = ac(h) forall h ∈ S. In particular f σ = a f .

Proof. We obviously have f σ ∈Mka(Qab). We consider g := a f − f σ ∈Mka(Qab). We note that the formg has non-zero Fourier coefficients only for h ∈ S with det(h) = 0. But then by [23, Proposition 6.16] wehave that g = 0. ut

We now want to consider the action of Gal(Qab/Q) on the Eisenstein series. We first consider the holo-morphic ones. That is, we consider the following two Eisenstein series

1. D(z,k/2;ka,χ,c) ∈ πβMka(Qab) for k ≥ n+12 ,

2. D(z,µ/2;ka,χ,c) ∈ πβMka(Qab) for k := n+1−µ and µ ≤ n+12 ,

where β is determined by Theorem 1. Note here that we take the field of definition to be Qab, i.e. theextension Φ does not appear. For this we refer to [23, Theorem 17.7].

In the following lemma we collect some properties that we will need concerning the functions Ξ(y,h;α,β )=

∏v∈a ξ (y,h;α,β ).

Lemma 3. Let h ∈ S with det(h) 6= 0 and y ∈ Sa+(R). Then we have for k ∈ 1

2Z we have

Ξ(y,h;k,0) = 2d(1−(n+1)/2)i−dnk(2π)dnkΓn(k)−dN(det(h))k−(n+1)/2en(iyh)

and for µ := n+1− k we have

Ξ(y,h;(n+1)/2,(µ− k)/2) = i−nk2−(dn(µ−k))/2π

dn(n+1)/2Γn(

n+12

)−d×

∏v∈a

det(yv)−( µ−k

2 )en(iyh)

Proof. The first statement is in [23, Equation 17.12]. For the second we have Ξ(y,h;(n+ 1)/2,µ/2−k/2) = ∏v∈a ξ (yv,hv;(n+1)/2,µ/2−k/2), where the function ξ (·) is given in [23, page 140]. By Shimura[19, Equation 4.35K] we have that ω(2πyv,hv;(n+ 1)/2,µ/2− k/2) = 2−n(n+1)/2ev(iyvhv). We concludethat

ξ (yv,hv;(n+1)/2,µ/2− k/2) = i−nk2−(n(µ−k))/2π

n(n+1)/2Γn(

n+12

)−1×

det(yv)−( µ−k

2 )ev(iyvhv),


where we have used the fact that δ−(hvyv) = 1 (the product of the negative eigenvalues of hvyv). Indeed wehave that δ−(hvyv) = δ−(y

1/2v hv

ty1/2v ). But the last quantity has the same number of negative eigenvalues as

the matrix hv, but hv > 0. ut

We will need the following Theorem (for a proof see [22, Theorem A6.5]).

Theorem 2. Let F be a totally real field, and let ψ be a Hecke character of F with ψa(b) = ∏v∈a

(bv|bv|

)k,

with 0 < k ∈ Z. For any integral ideal c of F put

Pc(k,ψ) := g(ψ)−1(2πi)−kd |DF |1/2Lc(k,ψ),

where d = [F : Q] and g(ψ) is a Gauss sum (defined [22, page 240]). Then Pc(k,ψ) ∈Q(ψ) and for everyσ ∈ Gal(Q(ψ)/Q) we have

Pc(k,ψ)σ = Pc(k,ψσ ).

We also summarize in the following lemma some more properties of Gauss sums.

Lemma 4. Let χ and ψ be two finite order Hecke characters of F and σ ∈ Gal(Qab/Q) we have

1. g(χ)σ = χ∗(qg)−1g(χσ ) where 0 < q ∈ Z so that e(1/N(f))σ = e(q/N(f)) , where f denotes the con-ductor of χ .

2.(

g(χψ)g(χ)g(ψ)

)σ

= g(χσ ψσ )g(χσ )g(ψσ ) .

3. If χ is a quadratic character then g(χ) = imN(f)1/2 where m is the number of archimedean primeswhere χv 6= 1.

We remark here that if we pick an element t ∈ Z×h so that e[t,Q]h = eh(t−1x) for x ∈ Q/Z then we have

that we can pick the q ∈ Z above so that rtp− 1 ∈ N(f)Zp for every prime p. Then we also obtain thatχ∗(qq) = χf(t).

3.3 Eisenstein series of integral weight.

We first consider the integral weight case. As we mentioned in the introduction these results can be found ina slightly different form in the book of Feit [8]. We also mention that results of this kind have been obtainedby Siegel, Harris, and Sturm, at least in the absolute convergence case. For completeness, and because ofsome notation and normalization issues, we partly reproduce these results here.

We start with the following proposition.

Proposition 4. For the Eisenstein series

D(z,k/2;ka,χ,c) = Lc(k,χ)[n/2]

∏i=1

Lc(2k−2i,χ2)E(z,k/2;ka,χ,c)

with k ≥ n+12 we have that π−β D(z,k/2;ka,χ,c) ∈Mka(Qab) and for all σ ∈ Gal(Qab/Q) we have that(

D(z,k/2;ka,χ,c)πβ P(χ)

)σ

=D(z,k/2;ka,χσ ,c)

πβ P(χσ ), σ ∈ Gal(Qab/Q),

where β = kd +∑[n/2]i=1 (2k− 2i) and P(χ) := g(χ)(i)kd

|DF |1/2

(∏

[n/2]i=1 (i)(2k−2i)d

)g(χ2[n/2])

|DF |b(n), with b(n) = 1/2 if [n/2] odd

and 1 otherwise.

Proof. We observe that we have that 2k−2i > 0 for all i = 1 . . . [n/2]. By definition we have that χa(b) =

∏v∈a

(bv|bv|

)k. By Theorem 2 above we have for


A(χ) :=|DF |1/2Lc(k,χ)

g(χ)(2πi)kd

[n/2]

∏i=1

|DF |1/2Lc(2k−2i,χ2)

g(χ2)(2πi)(2k−2i)d∈Qab

and for all σ ∈ Gal(Qab/Q) we have A(χ)σ = A(χσ ). Using Lemma 4 we may define the quantity

B(χ) =|DF |1/2Lc(k,χ)

g(χ)(2πi)kd

([n/2]

∏i=1

Lc(2k−2i,χ2)

(2πi)(2k−2i)d

)|DF |b(n)

g(χ2[n/2]),

where b(n) = 1/2 if [n/2] is odd and 1 otherwise. Then we have B(χ)σ = B(χσ ). By [8, Theorem 15.1] wehave E(z,k/2;ka,χ,c)σ = E(z,k/2;ka,χσ ,c) for all σ ∈ Gal(Q(χ)/Q). In particular we conclude that(

D(z,k/2;ka,χ,c)πβ P(χ)

)σ

=D(z,k/2;ka,χσ ,c)

πβ P(χσ ), σ ∈ Gal(Qab/Q),

where β = kd +∑[n/2]i=1 (2k−2i) and P(χ) := g(χ)(i)kd

|DF |1/2

(∏

[n/2]i=1 (i)(2k−2i)d

)g(χ2[n/2])

|DF |b(n). ut

Now we turn to the Eisenstein series

D(z,µ/2;ka,χ,c) = Lc(µ,χ)[n/2]

∏i=1

Lc(2µ−2i,χ2)E(z,µ/2;ka,χ,c),

and

D∗(z,µ/2;ka,χ,c) = Lc(µ,χ)[n/2]

∏i=1

Lc(2µ−2i,χ2)E∗(z,µ/2;ka,χ,c),

where we take µ ≤ n+12 , and k = n+1−µ .

We now prove

Lemma 5. Let β ∈ N as in Theorem 1 so that π−β D(z,µ/2;ka,χ,c) ∈Mka(Qab). Then we have that alsoπ−β D∗(z,µ/2;ka,χ,c) ∈Mka(Qab). Moreover for every σ ∈ Gal(Qab/Q) we have the reciprocity law(

D∗(z,µ/2;ka,χ,c)πβ i−dnk|DF |−nµ+3n(n+1)/4

)σ

=D∗(z,µ/2;ka,χσ ,c)

πβ i−dnk|DF |−nµ+3n(n+1)/4 .

Proof. The first statement i.e. that π−β D∗(z,µ/2;ka,χ,c) ∈Mka(Qab) follows from [23, Lemma 10.10].Moreover by Lemma 2 it is enough to establish the action of Gal(Qab/Q) on the full rank coefficients. ByProposition 2 and Lemma 3 we have that the hth Fourier coefficient c(h,χ) of π−β D∗(z,µ/2;ka,χ,c) withdet(h) 6= 0 is equal to

i−dnk2−(dn(µ−k))/2n

∏j=0

Γ (n+1

2− j/2)−d |DF |−nµ+3n(n+1)/4N(bc)−n(n+1)/2×

∏v∈c

fh,v (χ(πv)|πv|µ)×

Lc(µ−n/2,χρh), n even ;1, n odd.

If n is odd we have (c(h,χ)

i−dnk|DF |−nµ+3n(n+1)/4

)σ

=c(h,χσ )

i−dnk|DF |−nµ+3n(n+1)/4 .

Now we take n = 2m even. The character χρh has infinity type (χρh)a(b) = ∏v∈a

(bv|bv|

)1−µ+msince the

character ρh is the non-trivial character of the extension F(c1/2)/F with c := (−1)mdet(2h) and det(h)>>0 as h is positive definite for all real embeddings of F . Since 1−µ +m > 0 we have by [23, Theorem 18.12]that L(1−(1−µ +m),(χρh)

σ ) = L(1−(1−µ +m),(χρh))σ for all σ ∈Gal(Qab/Q). Hence we conclude

also in the case of n even that


(c(h,χ)

i−dnk|DF |−nµ+3n(n+1)/4

)σ

=c(h,χσ )

i−dnk|DF |−nµ+3n(n+1)/4 .ut

We now prove the following lemma

Lemma 6. Assume that (π−β D∗(z,µ/2;ka,χ,c))σ = aπ−β D∗(z,µ/2;ka,χσ ,c) for σ ∈ Gal(Qab/Q) a ∈Q×ab. Then

(π−β D(z,µ/2;ka,χ,c))σ = bπ−β D(z,µ/2;ka,χσ ,c)

where b = χ(qg)−na, where 0 < q ∈ Z such that e(1/N(c))σ = e(q/N(c)).

Proof. We use an argument due to Feit [8] and Sturm [25, Lemma 5] first introduced by Shimura in the caseof n = 1. We will need the reciprocity law of the action of the group G+×Gal(Q/Q) defined by Shimura in[23, Theorem 10.2]. We use the notation of Shimura in this theorem. Let t be an idele of F and as in Shimura

we define ı(t) :=(

1 00 t−1

). For a σ ∈Gal(Qab/Q) we define the element (ı(t),σ)∈G+×Gal(Q/Q) where

t ∈ Z×h corresponds to σ by class field theory and we extend σ to an element of the absolute Galois group.Moreover we may consider also ζh ∈ SpA as an element of G+×Gal(Q/Q) by taking (ζh,1). Then we havethat

(ı(t),σ)(ζh,1)(ı(t−1),σ−1)(ζ−1h ,1) =

((t 00 t−1

),1)

In particular we haveπ−β D(z,µ/2;ka,χ,c)(ı(t),σ)(ζh,1)(ı(t−1),σ−1)(ζ−1

h ,1) =

π−β D(z,µ/2;ka,χ,c)|k

(t 00 t−1

)= χc(t)n

π−β D(z,µ/2;ka,χ,c)

But then(π−β D(z,µ/2;ka,χ,c))σ = π

−β D(z,µ/2;ka,χ,c)(ı(t),σ) =

π−β D(z,µ/2;ka,χ,c)(ı(t),σ)(ζh,1)(ı(t−1),σ−1)(ζ−1

h ,1)(ζh,1)(ı(t),σ)(ζ−1h ),1) =

χc(t)n((

π−β D(z,µ/2;ka,χ,c)|k(ζh

)σ)|kζ−1h ) = χc(t)naπ

−β D(z,µ/2;ka,χσ ,c). (4)

ut

We can now establish the following corollary

Corollary 1. For the Eisenstein series D(z,µ/2;ka,χ,D) we have(D(z,µ/2;ka,χ,c)

πβ g(χn)i−dnk|DF |−nµ+3n(n+1)/4

)σ

=D(z,µ/2;ka,χσ ,c)

πβ g((χn)σ )i−dnk|DF |−nµ+3n(n+1)/4 ,

for all σ ∈ Gal(Qab/Q).

Proof. This follows immediately by combining Lemma 4 ((i) and (ii)), and the last two lemmas. ut

3.4 Eisenstein series of half integral weight.

Now we consider the case of half-integral weight. We will need the theta series θ(z) := ∑a∈gn ea(taza/2) ∈

M 12 a(Q,φ), where the quadratic character φ of Γ θ is defined by hγ(z) = φ(γ) ja

γ (z) for γ ∈ Γ θ . Note thatthis is the series θF defined in [23, page 39, equation 6.16] by taking in the equation there, using Shimura’snotation, u = 0 and λ the characteristic function of gn ⊂ Fn. Note in particular that since we are takingu = 0 we have that φF = θF . In particular Theorem 6.8 in (loc. cit) gives the properties of the series θ . Wenow prove the following lemma.

Lemma 7. For the theta series θ(z) and for σ ∈ Gal(Q/Q) we have that(θ | 1

2a(ζh

)σ

| 12 aζ−1h = θ


Proof. This follows immediately after observing that ζh ∈ Cθ and from Theorem 6.8 (4) in [23]. Indeedsince θ is invariant under Γ θ =G∩Cθ we have that θ | 1

2 aζh = θ 12 aζ−1h = θ . Since θ ∈M 1

2 a(Q), we concludethe proof. ut

Proposition 5. Let λ be equal to k or µ . Let β (λ ) ∈ N so that

π−β (λ )D∗(z,λ/2;ka,χ,c) ∈Mka(Qab).

Let σ ∈ Gal(Qab/Q and assume(π−β (λ )D∗(z,λ/2;ka,χ,c)

)σ

= α(λ )π−β (λ )D∗(z,λ/2;ka,χσ ,c), k = n+1−µ,

for some α(λ ) ∈Q×. Then we have π−β (λ )D(z,λ/2;ka,χ,c) ∈Mka(Qab) and(π−β (λ )D(z,λ/2;ka,χ,c)

)σ

= βπ−β (λ )D(z,λ/2;ka,χσ ,c)

where β = (χφ)c(t)nα(λ ).

Proof. The fact that π−β (λ )D(z,λ/2;ka,χ,c) ∈Mk(Qab) follows from [23, Lemma 10.10]. The rest of theproof was inspired by the proof of Theorem 10.7 in [23]. We write D(χ,λ ) for π−β (λ )D(z,λ/2;ka,χ,c). Letk′ = k+ 1

2 ∈Z. Then we note that θD(χ,λ )∈Mk′a(Qab) and for a σ ∈Gal(Qab/Q) we have θD(χ,λ )σ =(θD(χ,λ ))σ . Since θD(χ) is of integral weight we can apply the reciprocity-laws as before. Writing t ∈Z×hcorresponding to σ we have

(θD(χ,λ ))σ = (θD(χ,λ ))((ı(t),σ)(ζh,1)(ı(t−1),σ−1)(ζ−1h ,1))((ζh,1)(ı(t)σ)(ζ−1

h ,1))

=

((θD(χ,λ ))|k′

(t 00 t−1

))(ζh,1)(ı(t),σ)(ζ−1h ,1)

= (φ χ)c(t)n ((θD(χ)|k′aζh)σ )k′a ζ

−1h =

(φ χ)c(t)n(

φ(ζ0)(

θ | 12 aζh

)σ

(D(χ,λ )|kaζh)σ)|k′aζ

−1h =

φ(ζ0)φ(ζ0)−1((

θ | 12 aζh

)σ

| 12 aζ−1h

)((D(χ,λ )|kaζh)

σ)|kaζ

−1h =

(φ χ)c(t)nα(λ )θD(χσ ,λ ).

For the element ζ0 we refer to [23, page 132]. The last equation follows from the last Lemma. However theprevious equations deserve a comment. Note that for f1, f2 ∈M 1

2 a and γ ∈ Γ θ we have that ( f1 f2)|aγ =

φ(γ)( f1| 12 aγ)( f2| 1

2 aγ) since hγ(z)2 = φ(γ) jaγ (z).

So we obtain that θD(χ,λ )σ = (φ χ)c(t)−nαθD(χσ ). Since θ is not a zero divisor in the formal ring ofthe Fourier-expansion (see [23, page 74]) we conclude the proof. ut

We now establish also in the case of half-integral weight that

Proposition 6. Let β1 ∈ N so that π−β1D∗(z,k/2;ka,χ,c) ∈Mka(Qab). Let σ ∈ Gal(Qab/Q) Then for neven we have (

π−β1D∗(z,k/2;ka,χ,c)i−dnkC|DF |nk/2+3n(n+1)/4

)σ

=π−β1D∗(z,k/2;ka,χσ )

i−dnkC|DF |nk/2+3n(n+1)/4

and for n odd (π−β1D∗(z,k/2;ka,χ,c)

i−dnkC|DF |nk/2+3n(n+1)/4g(χ)|DF |1/2(2i)−(k−n)db([n/2])

)σ

=

π−β1D∗(z,k/2;ka,χσ )

i−dnkC|DF |nk/2+3n(n+1)/4g(χσ )|DF |1/2(2i)−(k−n)db([n/2]),

where b(m) = id if m is m is odd and 1 otherwise.


Let now β2 ∈ N so that π−β2D∗(z,µ/2;ka,χ,c) ∈Mka(Qab). Then we have(π−β2D∗(z,µ/2;ka,χ,c)

i−dnkC|DF |−n(n+1−k)+3n(n+1)/4

)σ

=π−β2D∗(z,µ/2;ka,χσ ,c)

i−dnkC|DF |−n(n+1−k)+3n(n+1)/4 ,

where k = n+1−µ .

Proof. Arguing as before, it is enough to consider the action of σ on the full rank coefficients. We consideran h with det(h) 6= 0. Then we have that the hth Fourier coefficient c(h,χ) of π−β1D∗(z,k/2;ka,χ,c) isequal to

2d(nk+1−(n+1)/2)i−dnk

(n−1

∏j=0

Γ (k− j/2)

)−d

N(det(h))k−(n+1)/2C|DF |nk/2+3n(n+1)/4×

N(bc)−n(n+1)/2∏v∈c

fh,v

(χ(πv)|πv|k+1/2

)×

π−d(k−n/2)Lc(k−n/2,χρh), n odd ;1, n even.

We now note that if n is even we have that k− (n+1)/2 ∈ Z and hence N(det(h))k−(n+1)/2 ∈Q×. Thenwe conclude that (

c(h,χ)i−dnkC|DF |nk/2+3n(n+1)/4

)σ

=c(h,χσ )

i−dnkC|DF |nk/2+3n(n+1)/4 .

In the case where n is odd we have that

Pc(k−n/2,χρh)σ = Pc(k−n/2,χσ

ρh), ∀σ ∈ Gal(Qab/Q),

withPc(k−n/2,χρh) := g(χρh)

−1(2πi)−(k−n/2)d |DF |1/2Lc(k−n/2,χρh)

We have g(χρh)σ

g(χσ ρh)= g(χ)σ g(ρh)

σ

g(χσ )g(ρh). Moreover we have that

g(ρh)σ

g(ρh)=

√

N(2det(h))σ

√N(2det(h))

, if [n/2] even;(iσi

)d √N(2det(h))σ

√N(2det(h))

, otherwise.

In particular since det(h) ∈ F+ we have(√N(2det(h))−1g(ρh)

)σ

√N(2det(h))−1g(ρh)

=

1, if [n/2] even;(

iσi

)d, otherwise.

For n odd we have that k− (n+1)/2 is half integral. Hence we conclude that(c(h,χ)

i−dnkC|DF |nk/2+3n(n+1)/4g(χ)|DF |1/2(2i)−(k−n)db([n/2])

)σ

=

c(h,χσ )

i−dnkC|DF |nk/2+3n(n+1)/4g(χσ )|DF |1/2(2i)−(k−n)db([n/2]),

where b(i) = id if [n/2] odd and 1 otherwise.Now we turn to the Eisenstein series D∗(z,µ/2;ka,χ,c). The Fourier coefficient c(h,χ) of π−β2D∗(z,µ/2;ka,χ,c)

for det(h) 6= 0 is equal to

i−dnk2−dn(µ−k)/2C|DF |−n(n+1−k)+3n(n+1)/4n−1

∏j=0

Γ (n+1

2− j/2)−dN(bc)−n(n+1)/2×

∏v∈c

fh,v

(χ(πv)|πv|n+1−k+1/2

)×

Lc(n/2+1− k,χρh), n odd ;1, n even.


Since we are taking k ≥ n+12 we have that Lc(n/2+1−k,χρh) ∈Q. Hence after observing that n+1−k+

1/2 ∈ Z we conclude that(c(h,χ

i−dnkC|DF |−n(n+1−k)+3n(n+1)/4

)σ

=c(h,χσ

i−dnkC|DF |−n(n+1−k)+3n(n+1)/4 ut

We can now conclude

Proposition 7. Let β1 ∈ N so that π−β1D(z,k/2;ka,χ,c) ∈Mka(Qab). Let σ ∈ Gal(Qab/Q) Then for neven we have (

π−β1D(z,k/2;ka,χ,c)g(χφ)ni−dnkC|DF |nk/2+3n(n+1)/4

)σ

=π−β1D(z,k/2;ka,χσ )

g(χσ φ)ni−dnkC|DF |nk/2+3n(n+1)/4

and for n odd (π−β1D(z,k/2;ka,χ,c)

g((χφ)n)i−dnkC|DF |nk/2+3n(n+1)/4g(χ)|DF |1/2(2i)−(k−n)db([n/2])

)σ

=

π−β1D(z,k/2;ka,χσ )

g((χn)σ φ)i−dnkC|DF |nk/2+3n(n+1)/4g(χσ )|DF |1/2(2i)−(k−n)db([n/2]),

where b(m) = id if m is m is odd and 1 otherwise.Let now β2 ∈ N so that π−β2D∗(z,µ/2;ka,χ,c) ∈Mka(Qab). Then we have(

π−β2D(z,µ/2;ka,χ,c)g(χφ n)i−dnkC|DF |−n(n+1−k)+3n(n+1)/4

)σ

=

π−β2D(z,µ/2;ka,χσ ,c)

g(((χφ)n)σ )i−dnkC|DF |−n(n+1−k)+3n(n+1)/4 ,

We now remark that the above proposition and Lemma 1 give a complete description of the reciprocitylaws of the Eisenstein series which we are considering. We summarize all the above in the following Theo-rem.

Theorem 3. Let k ∈ 12Z

a with kv ≥ (n+1)/2 for every v ∈ a. Let µ ∈ 12Z such that n+1−kv ≤ µ ≤ kv and

|µ− (n+1)/2|+(n+1)/2− kv ∈ 2Z for all v ∈ a. Then with a β ∈ N as in Theorem 1 we have

π−β D(z,µ/2;k,χ,c) ∈Nr

k(ΦQab),

and for every σ ∈ Gal(ΦQab/Φ) we have(π−β D(z,µ/2;k,χ,c)

ω(χ)

)σ

=π−β D(z,µ/2;k,χσ ,c)

ω(χσ ),

where ω(χ) is given as follows:

1. if k ∈ Za, µ ≥ (n+1)/2:

ω(χ) = in|p|g(χ)iµd+2µ[n/2]−[n/2]([n/2]+1)d |DF |−b(n)g(χ2[n/2]),

where p := (k−µa)2 and b(n) = 0 if [n/2] odd and 1/2 otherwise.

2. if k ∈ Za, µ < (n+1)/2:

ω(χ) = in|p|g(χn)i−dnν |DF |−nµ+3n(n+1)/4,

where ν := n+1−µ and p := k−νa2 .

3. if k 6∈ Za and µ ≥ (n+1)/2:


a. if n is evenω(χ) = in|p|g(χn)i−dnkC|DF |nk/2+3n(n+1)/4,

b. if n is odd

ω(χ) = in|p|g(χnφ)i−dnkC|DF |nµ/2+3n(n+1)/4g(χ)|DF |1/2(2i)−(µ−n)db([n/2]),

where p := (k−µa)2 and b(m) = id if m is m is odd and 1 otherwise and

4. if k 6∈ Za and µ < (n+1)/2:

ω(χ) = in|p|g(χφ)ni−dnνC|DF |−n(n+1−ν)+3n(n+1)/4,

where ν := n+1−µ and p := k−νa2 .

In particular we have thatπ−β D(z,µ/2;k,χ,c)

ω(χ)∈Nr

k(Φ(χ)),

where Φ(χ) is the finite extension of Φ obtained by adjoining the values of the character χ .

4 The L-function attached to a Siegel Modular Form

We start by discussing the Hecke algebras that we consider in this work. We follow closely Chapter V in[23]. As before we fix a fractional ideal b of F and an integral ideal c. We write C for D[b−1,bc] Moreoverwe define

E = ∏v∈h

GLn(gv), B = x ∈ GLn(F)h|x≺ g, X=CQC, Q = diag[r,r]|r ∈ B.

We write R(C,X) for the Hecke algebra corresponding to the pair (C,X) and for every place v ∈ h we wewrite R(Cv,Xv) for the local Hecke algebra at v and hence R(C,X) =

⊗vR(Cv,Xv). We now consider the

formal Dirichlet series with coefficients in the global Hecke algebra defined by T = ∑C\X/C CξC[νb(ξ )]and its local version at v ∈ h defined as Tv = ∑Cv\Xv/Cv CvξCv[νb(ξ )]. Here νb(ξ ) is defined by det(q)gwhere q ∈ B such that ξ ∈ D[b−1,b]diag[q−1,q∗]D[b−1,b]. We have that T= ∏vTv. Moreover if we definefor an integral g-ideal a the elements T (a) ∈R(C,X) and Tv(a) ∈R(Cv,Xv) for v ∈ h by

T (a) = ∑ξ∈X,νb(ξ )=a

CξC, Tv(a) = ∑ξ∈Xv,νb(ξ )=a

CvξCv

then we have that T = ∑a T (a)[a]. For an element f ∈Mk(C,ψ) we have an action of the Hecke algebraR(C,ψ) (see [22]). We denote this action by f|CξC for an element CξC ∈R(C,ψ). Assume now that forsuch an f 6= 0 we have f |T (a) = λ (a)f with λ (a) ∈ C for all integral g-ideals. Then Shimura shows thatthere exists λv,i ∈ C such that

L ·∑a

λ (a)[a] = ∏v∈h

Zv,

where the factors Zv are given by

Zv =

(1−N(p)n[p])−1

∏ni=1

((1−N(p)nλv,i[p])(1−N(p)nλ

−1v,i [p])

)−1, if v - c;

∏ni=1(1−N(p)nλv,i[p])

−1, otherwise.

and L := ∏p-c(1− [p])∏ni=1(1−N(p)2i[p]2

)−1, where the product is over the prime g-ideals prime to c. Fora Hecke character χ of F of conductor f we put

Z(s, f,χ) := ∏v∈h

Zv(χ∗(q)N(q)−s) , (5)


where Zv (χ∗(q)N(q)−s) is obtained from Zv by substituting χ∗(p)N(p)−s for [p]. We will need another

L-function which we will denote by Z′(s, f,χ) and we define by

Z′(s, f,χ) := ∏v∈h

Zv(χ∗(q)(ψ/ψc)(πq)N(q)−s) , (6)

where πq a uniformizer of Fq. We note here that we may obtain the first from the second up to a finitenumber of Euler factors by setting χψ−1 for χ .

5 The Rankin-Selberg Method

We now explain the integral representation of the zeta function introduced above due to Shimura. Everythingin this section is taken from [23, paragraph 20 and 22] as well as [20].

We write L for the set of all g-lattices in Fn1 . We set L0 := gn

1 and we remark that for an element L ∈ L

we can find an element y ∈ GLn(F)h such that L = yL0. For an element τ ∈ S we define

Lτ := L ∈ L|`∗τ` ∈ bd−1, ∀` ∈ L.

Let f∈Mk(C,ψ), τ ∈ S+ and q∈GLn(F)h. Following Shimura we define the following two formal Dirichletseries

D(τ,q; f) := ∑x∈B/E

ψc(det(qx))|det(x)|−n−1F c(τ,qx; f)[det(x)g], (7)

andD′(τ,q; f) := ∑

x∈B/Eψ(det(qx))|det(x)|−n−1

F c(τ,qx; f)[det(x)g]. (8)

We note that the second is obtained from the first one by setting (ψ/ψc)(t)[tg] for [tg], t ∈ F×h in D(τ,q; f)and multiplying by (ψ/ψc)(det(q)). We define the series

L0 = ∏v-c

([(n+1)/2]

∏i=1

(1−N(p)2n+2−2i[p]2)

)−1

.

Then we have

Theorem 4 (Shimura). Let 0 6= f ∈Mk(C,ψ)) and such that f|T (a) = λ (a)f for every a. Then for τ ∈S+∩GLn(F) and L = qL0 with q ∈ GLn(F)h we have

D(τ,q; f)L0 ∏v∈b

gv[p]∏v-c

hv([p])−1 =

∏v∈h

Zv ∑L<M∈Lτ

µ(M/L)ψc(det(y))[det(q∗yg]c(τ,y; f).

Assume now that kv ≥ n/2 for some v ∈ a. Then there exists τ ∈ S+∩GLn(F) and r ∈ GLn(F)h such that

0 6= ψc(det(r))cf(τ,r)∏v∈h

Zv = D(τ,r; f) ·L0 ∏v-c

hv([p])−1 ·∏

v∈bgv([p]).

For the definitions of gv(·) and hv(·) we refer to [23]. Now given a Hecke character χ of F , τ ∈ S+ andr ∈ GLn(F)h we define a Dirichlet series as follows:

D′r,τ(s, f,χ) := ∑B/E

ψ(det(rx))χ∗(det(x)g)cf(τ,rx)|det(x)|s−n−1F . (9)

This series is obtained from the series in Equation 8 by putting χ∗(tg)|tg|sF for [tg]. In particular we havethe equation


D′r,τ(s, f,χ)Λc

(2s−n

4

)∏v∈b

gv(χ(ψ/ψc)(πv)|πv|s) = (10)

Z′(s, f,χ)(ψ/ψc)2(det(r))×

∑L<M∈Lτ

µ(M/L)(ψ2c/ψ)(det(y))χ∗(det(r∗y)g)|det(r∗y|sF c(τ,y; f),

where for an integral ideal a we write

Λa(s) =

La(2s,ρτ ψχ)∏

n/2i=1 La(4s−2i,ψ2χ2), if n is even;

∏(n+1)/2i=1 La(4s−2i+1,ψ2χ2), n is odd.

,

Given χ as above we write f for the conductor of χ . We define t ′ ∈ Za by

(ψχ)a(x) = x−t ′a |xa|t

′.

and µ ∈ Za by the conditions 0≤ µv ≤ 1 for all v ∈ a and µ− [k]− t ′ ∈ 2Za.We now define a weight l and a Hecke character ψ ′ of F by l = µ +(n/2)a and ψ ′ = χ−1ρτ , where ρτ is

the Hecke character of F corresponding to the extension F(c12 )/F with c := (−1)[n/2]det(2τ). Let us write

θχ ∈Ml(C′,ψ ′) for the theta series associated to the data (χ−1,µ,τ,r) in section 2. Write C′=D[b′−1,b′c′]and define e := b+b′. Then we have (see [20, page 572]),

Theorem 5 (Shimura).

(4π)−n(su+(k+l)/2)(√

DF N(e)−1)n(n+1)/2∏v∈a

Γn(s+(kv + lv)/2)D′r,τ(2s+3n/2+1; f,χ) =

|det(r)|−2s−n/2F det(τ)+(k+µ+nu/2)/2+su×∫

Φ

f (z)θχ(z)E(z, s+(n+1)/2,k− l,εψχρτ ,Γ ′)δ (z)kdz,

where Φ :=H/Γ ′ and Γ ′ := G∩D[e−1,eh], where h = e−1(bc∩b′c′). here ε = 1 if n is even and it is thenon-trivial character of F((−1)

n2 )/F otherwise.

In particular using the equation 10 we obtain

Theorem 6 (Shimura).

Z′(s, f,χ)∏v∈a

Γn

(s−n−1+ kv +µv

2

)×

(ψ/ψc)2(det(r)) ∑

L<M∈Lτ

µ(M/L)(ψ2c/ψ)(det(y))χ∗(det(r∗y)g)|det(r∗y)|sF c(τ,y; f) =

(D−1/2

F N(e))n(n+1)/2

(4π)n||s′u+λ ||det(τ)s′u+λ |det(r)|n+1−sA ×

∏v∈b

gv((ψ/ψc)(πv)χ∗(πvg)|πv|s)(Λc/Λh)((2s−n)/4)vol(Φ)< f ,θχ D((2s−n)/4)>Γ ′ ,

where s′ = (2s−3n−2)/4 and for an integral ideal a of F,

Λa(s) =

La(2s,ρτ ψχ)∏

n/2i=1 La(4s−2i,ψ2χ2), if n is even;

∏(n+1)/2i=1 La(4s−2i+1,ψ2χ2), n is odd.

,

andD(s) = Λh(s)E(z, s;k− l,ε,ρτ ψχ,Γ ′).

We have normalized the Petersson inner product as follows

< f ,θχ D((2s−n)/4)>Γ ′=1

vol(Φ)

∫Φ

f (z)θχ(z)D(z,(2s−n)/4)δ (z)kdz.


In particular there exists (τ,r) with c(τ,r; f) 6= 0 such that

Z′(s, f,χ)∏v∈a

Γn

(s−n−1+ kv +µv

2

)ψc(det(r))c(τ,r; f) = (11)

(D−1/2

F N(e))n(n+1)/2

(4π)n||s′u+λ ||det(τ)s′u+λ |det(r)|n+1−sA ×

∏v∈b

gv((ψ/ψc)(πv)χ∗(πvg)|πv|s)(Λc/Λh)((2s−n)/4)vol(Φ)< f ,θχ D((2s−n)/4)>Γ ′ .

We note here that vol(Φ) ∈ πn(n+1)/2Q×.

6 Petersson Inner Products and Periods

In this section we define some archimedean periods that we will use to normalize the special values of thefunction Z′(s, f,ψ). The idea of defining these periods is due to Sturm [25] (building on previous work ofShimura), who considered the case of n even and F =Q. However one should notice also the difference onthe bounds of the weights that we impose. In what it follows we will call a Hecke operator T (a), relative tothe group C = D[b−1,bc], as “good” if a is prime to c

Theorem 7. Let f∈ Sk(c,ψ) be an eigenform for all the “good” Hecke oparators of C. Let Φ be the Galoisclosure of F over Q and write Ψ for extension of Φ generated by the eigenvalues of f and their complexconjugation . Assume m0 := minv(kv) > [3n/2+ 1] + 2. Then there exists a period Ωf such that for anyg ∈ Sk(Q) we have (

< f,g >

Ωf

)σ

=< fσ ,gσ ′ >

Ωfσ

,

for all σ ∈ Gal(Q/Φ), where σ ′ = ρσρ . Moreover Ωf depends only on the eigenvalues of f and we have<f,f>

Ωf∈Ψ×.

Remark 1. As we remarked above, a theorem of this form has been firstly proved by Sturm [25], whenF = Q and n is even. A similar theorem appears also in the work of Panchishkin [17]. It is also importantto notice that in Panchishkin’s theorem one can take also g not cuspidal. However for this he has to takethe weight big enough in order to be in the range of absolute convergent for the Eisenstein series (see theTheorems after the proof). Our proof is modelled on that of Sturm [25, Theorem 3] and of Shimura [23,Theorem 28.5]. Maybe one should here remark that one of the differences with the proof here in comparisonwith the one of Sturm is that we use the identity (10) and not the Andrianov-Kalinin identity used by Sturm.Finally since we are using a stronger theorem of Shimura with respect to the absolute convergence of thefunction Z(s, f,χ) we also obtain better bounds for the weights. Finally we remark the slightly larger boundon m0 than in Shimura [23, Theorem 28.5]. The reason for this is the above mentioned problem with theEisenstein spectrum (i.e. separate it rationally from the cuspidal part).

Proof. We write λ (a) for the system of the eigenvalues of f (with respect to the “good” Hecke operators)and we define V := h ∈ Sk(c,ψ)|h|T (a) = λ (a)h. Then as in Shimura we define V(Ψ) = V∩Sk(c,ψ;Ψ).By [10] we have that the space V(Ψ) is preserved by the operators T (a). Moreover the “good” Heckeoperators generate a ring of semi-simple Ψ -linear transformations hence we have V = V(Ψ)⊗Ψ C andSk(C,Ψ) = V(Ψ)⊕U, with U a vector space over Ψ which is stable under the action of the “good” Heckeoperators. Since an eigenform in U⊗Ψ C which is not contained in V must be orthogonal to it we have thatthe above decomposition is orthogonal with respect to the Petersson inner product.

We now pick an integer σ0 so that 3n/2+ 1 < σ0 < m0 and m0−σ0 6∈ 2Z. Note that this is alwayspossible thanks to our assumption m0 > [3n/2+1]+2. Then we define µ ∈Za by the conditions 0≤ µv ≤ 1and σ0− kv + µv ∈ 2Z for all v ∈ a. Our choice of σ0 implies in particular that there exists an v ∈ a sothat µv 6= 0. We put t ′ := µ − k. We now pick a quadratic character χ of F so that (ψχ)a(x) = xt ′

a |xa|−t ′

and of conductor f such that c|f. Note that such a character can be obtained as the non trivial characterof the quadratic extension F(

√∆) by picking the sign of ∆ ∈ F properly at v ∈ a and ∆ with non trivial


valuation at all primes that divide c. The existence of such a ∆ follows from the approximation theorem forF . As in Shimura [23, page 236] we define l := µ +(n/2)a and ν = σ0− (n/2). Then ν ≥ (n+ 1)/2 and0≤ k− l−νa ∈ 2Za. We consider the theta series θχ with respect to our choices of χ and µ . By Theorem6, after evaluating at s = σ0 we obtain

Z′(σ0, f,χ)∏v∈a

Γn

(σ0−n−1+ kv +µv

2

)(ψ/ψc)

2(det(r))×

∑L<M∈Lτ

µ(M/L)(ψ2c/ψ)(det(y))χ∗(det(r∗y)g)|det(r∗y)|σ0

F c(τ,y; f) =

(D−1/2

F N(e))n(n+1)/2

(4π)n||s′0u+λ ||det(τ)s′0u+λ |det(r)|n+1−σ0A (Λc/Λh)(ν/2)×

∏v∈b

gv((ψ/ψc)(πv)χ∗(πvg)|πv|σ0)vol(Φ)< f ,θχ D(ν/2,ερτ ψχ)>Γ ′ ,

where s′0 = (2σ0−3n−2)/4. We now note (see [23, page 237]) that

∏v∈a Γn

(σ0−n−1+kv+µv

2

)vol(Φ)

∈ πn|| k−l−νa

2 ||−n||k||+dεQ×

where ε = n2/4 if n even and (n2−1)/4 otherwise. We now write δ for the rational part of∏v∈a Γn

(σ0−n−1+kv+µv

2

)vol(Φ) .

We now take β ∈ N so that π−β D(ν/2) ∈ Npk−l(ΦQab) with p = k−l−νa

2 and we set γ := n|| k−l−νa2 || −

n||k||+dε−n||s′0u+λ ||−β . We further set

B(χ,ψ,τ,r, f) := δ (ψ/ψc)2(det(r))×

∑L<M∈Lτ

µ(M/L)(ψ2c/ψ)(det(y))χ∗(det(r∗y)g)|det(r∗y)|σ0

F c(τ,y; f),

andC(χ,ψ,τ,r) := (N(e))n(n+1)/2 |det(r)|n+1−σ0

A ×

∏v∈b

gv((ψ/ψc)(πv)χ∗(πvg)|πv|σ0)(Λc/Λh)(ν/2).

We then have for every σ ∈ Gal(Q/Φ) that

B(χ,ψ,τ,r, f)σ = B(χσ ,ψσ ,τ,r, fσ ) and C(χ,ψ,τ,r)σ =C(χσ ,ψσ ,τ,r).

We now note the equalities< f ,θχ D(ν/2,ερτ ψχ)>Γ ′=

< f ,p(θχ D(ν/2,ερτ ψχ))>Γ ′=< f ,TrΓ

Γ ′(p(θχ D(ν/2,ερτ ψχ)))>Γ ,

where p :Rpk → Sk is Shimura’s holomorphic projection operators [23, Proposition 15.6](note that θχ D(ν/2)∈

Rpk since θχ is a cusp form) and TrΓ

Γ ′ : Sk(Γ′,ψ)→ Sk(Γ ,ψ) is the usual trace operator attached to the groups

Γ ′ ≤ Γ . Moreover, since θχ π−β D(ν/2) ∈Npk (ΦQab), we may consider the action of σ ∈ Gal(ΦQab/Φ).

Thenp(θχ π

−β D(ν/2,ερτ ψχ))σ = p(θ σχ (π−β D(ν/2,ερτ ψχ))σ ),

and,TrΓ

Γ ′(θχ π−β D(ν/2,ερτ ψχ))σ = TrΓ

Γ ′(θσχ D(ν/2,ερτ ψχ)σ ),

where in the last equation the last trace is from the space Sk(Γ′,ψσ ) to Sk(Γ

′,ψσ ). The equivariant propertyof the holomorphic projection operator is shown in Proposition 15.6 of (loc. cit.) and the one of the traceis exactly as in Sturm where he considers the case of F = Q, but the arguments is valid also for general Fsince the strong approximation theorem also hold for the group Spn(F), the essential argument in his proof.We make this more formal in the lemma following this proof.

Keeping now the character χ fixed we know that for any given f ∈ V there exists (τ,r) such that


B(χ,ψ,τ,r, f) = δψ(det(r))c(τ,r; f) 6= 0.

We note here that the same pair (τ,r) can be used for the form fσ , as it follows from the proof of Theorem20.9 in [23]. As in Shimura we write G for the set of pairs (τ,r) for which such an f exists. From theobservation above the set G is the same also for the system of eigenvalues λ (a)σ , for all σ ∈ Gal(Q/Φ).In particular for such an (τ,r)

0 6= πγ Z′(σ0, f,χ)δψ(det(r))c(τ,r; f) = (12)(

D−1/2F N(e)

)n(n+1)/2(4)n||s′0u+λ ||det(τ)s′0u+λ |det(r)|n+1−σ0

A ×

∏v∈b

gv((ψ/ψc)(πv)χ∗(πvg)|πv|σ0)(Λc/Λh)(ν/2)< f ,θχ π

−β D(ν/2,ερτ ψχ)>Γ ′ .

The fact that Z′(σ0, f,χ) 6= 0 is in principle [23, Theorem 20.13]. Indeed in page 183 of (loc. cit) Shimurafirst proves the non-vanishing of Z′(σ0, f,χ) for any character χ with µ 6= 0, as it is the case that we consider.Further we note that this in particular implies also that C(χ,ψ,τ,r) 6= 0 for all (τ,r) ∈G.

We now define an element gτ,r,ψ ∈ Sk(Γ ,ψ;ΦQab) by

gτ,r,ψ = π−β TrΓ

Γ ′

(p(θχ π

−β D(ν/2,ερτ ψχ))),

and define the space W to be the space generated by gτ,r,ψ for (τ,r) ∈G. We now consider the case n evenor odd separately.

The case of n even: In this case we have that ε is the trivial character. We now claim that there exists anΩf ∈ C× such that any f ∈ V and any gτ,r,ψ(

< f,gτ,r,ψ >

Ωf

)σ

=< fσ ,gσ ′

τ,r,ψ >

Ωfσ

,

where σ ′ = ρσρ . First we observe that

gσ ′τ,r,ψ = TrΓ

Γ ′

(p(θχ π

−β D(ν/2,ερτ ψχ)))σ ′

= TrΓ

Γ ′

(p(θ σ ′

χ (π−β D(ν/2,ερτ ψχ))σ ′)=

TrΓ

Γ ′

(p(θχ(π

−β D(ν/2,ερτ ψχ))σ ′),

where the last equality follows from the fact that χ is a quadratic character. We now recall that D(ν/2,ρτ ψχ)=D(z,ν/2;k− l,ρτ ψχ,Γ ) and we have seen that(

π−β D(z,ν/2;k− l,ρτ ψχ,Γ )

P(ρτ ψχ)

)σ ′

=π−β D(z,ν/2;k− l,ρτ ψχ

σ ′ ,Γ )

P(ρτ ψχσ ′)

,

where P(ρτ ψχ) = in|p|g(ρτ ψχ)(i)νd

|DF |1/2

(∏

[n/2]i=1 (i)(2ν−2i)d

)g(ψ2[n/2])

|DF |b(n), with p = k−l−νa

2 . We conclude that

gσ ′τ,r,ψ =

P(ρτ ψχ)σ ′

P(ρτ ψσ ′

χ)TrΓ

Γ ′

(p(θχ π

−β D(ν/2,ερτ ψσ ′

χ)))=

P(ρτ ψχ)σ ′

P(ρτ ψσ ′

χ)gτ,r,ψσ .

We set R(ψ) := in|p|(i)νd

|DF |1/2

(∏

[n/2]i=1 (i)(2ν−2i)d

)g(ψ2[n/2])

|DF |b(n). We now consider the ratio

g(ρτ ψχ)σ ′

g(ρτ ψσ ′

χ)=

g(ρτ)σ ′

g(ρσ ′τ )

g(ψ)σ ′

g(ψσ ′)

g(χ)σ ′

g(χσ ′).

We recall that ρτ is the non-trivial character of the quadratic extension F(√

c)/F with c = (−1)[n/2]det(2τ).Since we are considering τ > 0 we have that


g(ρτ)σ ′

g(ρσ ′τ )

=

√

2det(τ)σ ′

√2det(τ)

, if [n/2] even;(iσ′

i

)d √N(2det(τ))σ ′

√N(2det(τ))

, otherwise.

Putting all these together we conclude that

gσ ′τ,r,ψ =

g(ρτ)σ ′

g(ρσ ′τ )

g(ψ)σ ′

g(ψσ ′)

g(χ)σ ′

g(χσ ′)

R(ψ)σ ′

R(ψσ ′)gτ,r,ψσ

For any gτ,r we have

(4)−n||s′0u+λ ||Dn(n+1)/4F π

γ Z′(σ0, f,χ)B(χ,ψ,τ,r, f) =

det(τ)s′0u+λC(χ,ψ,τ,r)< f,gτ,r >Γ .

For any (τ,r) ∈G we have seen that C(χ,ψ,τ,r) 6= 0. We obtain

< f,gτ,r >Γ

(4π)−n||s′0u+λ ||Dn(n+1)/4F Z′(σ0, f,χ)

= det(τ)−(s′0u+λ ) B(χ,ψ,τ,r, f)

C(χ,ψ,τ,r).

For any σ ∈ Gal(Q/Q) we have then(< f,gτ,r,ψ >Γ

(4)−n||s′0u+λ ||Dn(n+1)/4F πγ Z′(σ0, f,χ)

)σ

=

(det(τ)−(s

′0u+λ ) B(χ,ψ,τ,r, f)

C(χ,ψ,τ,r)

)σ

=

(det(τ)−(s′0u+λ ))σ B(χσ ,ψσ ,τ,r, fσ )

C(χσ ,ψσ ,τ,r)=

(det(τ)−(s′0u+λ ))σ

det(τ)−(s′0u+λ )

< fσ ,gτ,r,ψσ >Γ

(4π)−n||s′0u+λ ||Dn(n+1)/4F Z′(σ0, fσ ,χ)

.

We remark that s′0u + λ = 2σ0−3n−24 u +

k+µ+ n2 u

2 = σ0u+k+µ

2 − n+12 u. By our choice of σ0 we have that

σ0u+ k+ µ ∈ 2Za. We obtain that (det(τ)−(s′0u+λ ))σ

det(τ)−(s′0u+λ )

= (det(τ)12 a)σ

det(τ)(12 a)

Now we note that since det(τ) is totally

positive we have (g(ρτ)σ ′

g(ρσ ′τ )

)−1(det(τ)

12 a)σ

det(τ)(12 a)

=

1, if [n/2] even;(

iσ′

i

)d, otherwise.

We have seen that

gτ,r,ψσ =

(g(ρτ)

σ ′

g(ρσ ′τ )

g(ψ)σ ′

g(ψσ ′)

g(χ)σ ′

g(χσ ′)

R(ψ)σ ′

R(ψσ ′)

)−1

gσ ′τ,r,ψ .

and hence (< f,gτ,r,ψ >Γ

(4)−n||s′0u+λ ||Dn(n+1)/4F πγ Z(σ0, f,χ)

)σ

=

(g(ρτ)σ ′

g(ρσ ′τ )

g(ψ)σ ′

g(ψσ ′)

g(χ)σ ′

g(χσ ′)

R(ψ)σ ′

R(ψσ ′)

)−1(det(τ)

12 a)σ

det(τ)(12 a)×

< fσ ,gσ ′τ,r,ψ >Γ

(4)−n||s′0u+λ ||Dn(n+1)/4F πγ Z′(σ0, fσ ,χ)

,


or equivalently (< f,gτ,r,ψ >Γ

(g(ψ)g(χ)R(ψ))−1 B(n)4−n||s′0u+λ ||Dn(n+1)/4F πγ Z′(σ0, f,χ)

)σ

=

< fσ ,gσ ′τ,r,ψ >Γ(

g(ψσ ′)g(χ)R(ψσ ′)B(n))−1

4−n||s′0u+λ ||Dn(n+1)/4F πγ Z′(σ0, fσ ,χ)

.

where B(n) = id if [n/2] is odd and 1 otherwise. Hence we define

Ωf := (g(ψ)g(χ)R(ψ))−1 B(n)4−n||s′0u+λ ||Dn(n+1)/4F π

γ Z′(σ0, f,χ)

The case of n odd: We now repeat the considerations above but with the half-integral weight Eisensteinseries. (

π−β D(z,ν/2;k− l,εχψρτ ,c)

g(ερτ ψχφ)nin|p|i−dnνC|DF |nν/2+3n(n+1)/4g(ερτ ψχ)|DF |1/2(2i)−(ν−n)db([n/2])

)σ ′

=

π−β D(z,ν/2;k− l,(εχψρτ)σ )

g((εχψρτ)σ ′φ)nin|p|i−dnνC|DF |nν/2+3n(n+1)/4g((εχψρτ)σ ′)|DF |1/2(2i)−(ν−n)db([n/2]),

where b(m) = id if m is m is odd and 1 otherwise. We set P(εχψρτ) equal to

g(ερτ ψχφ)nin|p|i−dnνC|DF |nν/2+3n(n+1)/4g(ερτ ψχ)|DF |1/2(2i)−(ν−n)db([n/2])

and as before we have

gσ ′τ,r,ψ =

P(ερτ ψχ)σ ′

P(ερτ ψσ ′

χ)gτ,r,ψσ .

We consider the ratio(g(ερτ ψχφ)ng(ερτ ψχ))σ ′

g(ερτ ψσ ′

χφ)ng(ερτ ψσ ′

χ)=

(g(ε)σ ′

g(ε)

)n+1(g(ρτ)

σ ′

g(ρτ)

)n+1(g(ψ)σ ′

g(ψσ ′)

)n+1(g(χ)σ ′

g(χ)

)n+1(g(φ)σ ′

g(φ)

)n

.

Since n+1 is even and ρτ ,χ,ε are quadratic characters we get that(g(ε)σ ′

g(ε)

)n+1

=

(g(ρτ)

σ ′

g(ρτ)

)n+1

=

(g(χ)σ ′

g(χ)

)n+1

= 1.

We set R := in|p|i−dnνC|DF |nν/2+3n(n+1)/4|DF |1/2(2i)−(ν−n)db([n/2]), and then we have

gσ ′τ,r,ψ =

(g(ψ)σ ′

g(ψσ ′)

)n+1(g(φ)σ ′

g(φ)

)nRσ ′

Rgτ,r,ψσ .

By the same calculations as in the case of n even, by no noticing that s′0u+λ ∈ Za we obtain For anyσ ∈ Gal(Q/Q) we have then(

< f,gτ,r,ψ >Γ

4−n||s′0u+λ ||Dn(n+1)/4F πγ Z′(σ0, f,χ)

)σ

=< fσ ,gτ,r,ψσ >Γ


.

Hence we conclude


< f,gτ,r,ψ >Γ(g(ψ)

n+1g(φ)

nR)−1

4−n||s′0u+λ ||Dn(n+1)/4F πγ Z′(σ0, f,χ)

σ

=

< fσ ,gτ,r,ψσ >Γ(g(ψσ ′)

n+1g(φ)

nR)−1


.

So for n odd we define

Ωf :=(

g(ψ)n+1

g(φ)nR)−1

4−n||s′0u+λ ||Dn(n+1)/4F π

γ Z′(σ0, f,χ).

By W′ we define the space generated by the projection of W on V. By definition W′ = V. Indeed forany element g ∈ V there exists h ∈W′ such that < g,h >Γ 6= 0, simply by taking the projection of thecorresponding gτ,r to W′. So the C span of gτ,r with τ,r ∈ G is equal to V. Since gτ,r have algebraiccoefficients we have that the Q-span is equal to V(Q). We can now establish the theorem for any g ∈ V(Q)since after writing g = ∑ j c jgτ j ,r j ,V ∈ V(Q), where gτ j ,r j ,V is the projection of gτ j ,r j to V, we have

(< f,g >

Ωf

)σ

= ∑j

c jσ

< fσ ,gσ ′τ j ,r j ,V

>

Ωfσ

=< fσ ,gσ ′ >

Ωfσ

We now take any g ∈ Sk(Γ ,ψ;Q). We write g = g1 +g2 with g1 ∈ V and 2 ∈ V⊥. Then we have that(< f,g >

Ωf

)σ

=

(< f,g1 >

Ωf

)σ

=< fσ ,gσ ′

1 >

Ωfσ

=< fσ ,gσ ′ >

Ωfσ

where the last equality follows from the fact that < f,g >= 0 implies that < fσ ,gσ ′ >= 0. It is enough toshow this for g an eigenform for all the good Hecke operators in an L-packet different from that of f’s. Thatis, there exists an ideal a with (a,c) = 1 so that T (a)f = λff and T (a)g = λgg such that λf 6= λg. But thenwe have

λσf < fσ ,gσ ′ >=< T (a)fσ ,gσ ′ >=

< fσ ,T (a)gσ ′ >=< fσ ,λ σ ′g gσ ′ >=< fσ ,gσ ′ > λ

σg

and hence we conclude that < fσ ,gσ ′ >= 0. Here we have used the facts that the good Hecke operators areself adjoint with respect to the Petersson inner produt, and that their Hecke eigenvalues are totally real (forboth facts see [23, Lemma 23.15]).

Finally taking g equal to f we obtain that Ωf is equal to < f, f > up to a non-zero element in the the Galoisclosure of the field generated by the Fourier coefficients of f (note that it also contains the eigenvalues). ut

We now give a proof of the equivariant property of the trace that we used in the proof of the theorem.The proof follows the proof given by Sturm [25, Lemma 11] extended to the totally real field situation.

Lemma 8. With notation as in the proof of the above theorem we have for any f ∈ Sk(Γ′,ψ;Qab)

TrΓ

Γ ′( f )σ = TrΓ

Γ ′( f σ ), σ ∈ Gal(ΦQab/Φ).

Proof. Thanks to the strong approximation for Spn(F) we may work adelically. We write D and D′ for thecorresponding to Γ and Γ ′ adelic groups (i.e. Γ = G∩D). We fix elements gi ⊂ Dh so that D =

⋃D′gi.

For t ∈ Z×h corresponding to σ |Qab we note that(1n 00 t−11n

)gi

(1n 00 t1n

)∈ Spn(A)h

and hence by strong approximation we can find elements ui ∈ D′ with f |ui = f (i.e. ψ(ui) = 1) andwi ∈ Spn(F) so that


(1n 00 t−11n

)gi

(1n 00 t1n

)= uiwi.

We moreover note that wia = ui−1a . Now we claim that since the gi’s form a set of representatives of the

classes of D′ in D, the same holds for(

1n 00 t−11n

)gi

(1n 00 t1n

), and hence also for wi since ui ∈D′. Indeed

since t ∈ Z×h → F×h we have that(1n 00 t−11n

)(a bc d

)(1n 00 t1n

)=

(a tb

t−1c d

)∈ D[a,b]

if(

a bc d

)∈ D[a,b], for some fractional ideals a,b with ab⊆ g. In particular we have that ı(t)giı(t−1) ∈ D.

We claim that D =⋃

iD′ı(t)giı(t−1). Indeed let d ∈ D. Then ı(t−1)dı(t) ∈ D and hence there exists d′ ∈ D′

such that ı(t−1)dı(t) = d′g j for some j. Or equivalently d = ı(t)d′g jı(t−1) = ı(t)d′ı(t−1)ı(t)g jı(t−1), whichestablishes our claim since ı(t)d′ı(t−1)ı(t) ∈ D′.

We now consider the elements (ı(t),σ),(wi, id),(gi, id) ∈ G+×Gal(Q/Q). Then we have

(TrΓ

Γ ′( f σ ))σ−1= (∑

iψ(gi)

σ f σ |kgi)σ−1

=

∑i(ψ(gi) f )((ı(t),σ)(gi,1)(ı(t−1),σ−1)) = ∑

iψ(gi) f (uiwi,1) = ∑

iψ(gi) f |kwi.

The proof of the lemma is now completed after observing that ψ(gi) = ψ(wi). ut

We also mention here the following theorem of Garrett [10].

Theorem 8 (Garrett). Let k > 2n+ 1 and f,g ∈ Ska. Take f an eigenform for almost all Hecke operators.Then for all σ ∈ Aut(C/Q), we have (

< fρ ,g >

< fρ , f >

)σ

=< fσρ ,gσ >

< fσρ , fσ >

In particular if we take f,g ∈ Ska(Q), and take f with totally real Fourier coefficients then we have that<fρ ,g><fρ ,f> ∈Q and (

< f,g >

< f, f >

)σ

=< fσ ,gσ >

< fσ , fσ >, σ ∈ Gal(Q/Q).

We note that if we combine the above result of Garrett with the following result of Harris on the Eisen-stein spectrum

Theorem 9 (Harris). Let k > 2n+ 1 and write Eka for the orthogonal complement of Ska in Mka (theEisenstein series). Define Eka(Q) :=Mka(Q)∩Eka. Then we have

Mka(Q) = Eka(Q)⊕Ska(Q).

Proof. This follows from the work of Harris [10]. Indeed in general we have that (see [23, Theorems 27.14,and 27.16])

Mka(Q) = Eka(Q)⊕Ska(Q)

and Eka(Q) =⊕nr=0E

rka(Q) where Er

ka the space of Klingen type Eisenstein series associated to a parabolicgroup Pr stabilizing an isotropic space of dimension r. Harris has shown that in the case of weight as above(i.e. the absolute convergence situation) we have that Er

ka(Q) = Erka(Q)⊗QQ. ut

Now this theorem allows us to take g ∈Mka in Theorem 8.


7 Algebraicity results for Siegel Modular Forms over totally real fields

In this section we present various results regarding special values of the function Z′(s, f,χ), with f ∈Sk(b,c,ψ), an eigenform for all Hecke operators. We remind that we have also considered the functionZ(s, f,χ). The two coincide when the Nebentypus of f is trivial. Indeed if we write Zv(χ

∗(πvg)|πv|s) for theEuler factor of Z(s, f,χ) at some prime v ∈ h then the corresponding Euler factor of Z′(s, f,χ) is equal toZv((ψ/ψc)χ

∗(πvg)|πv|s). We note the equation

Z′(s, f,χψ−1) = Zc(s, f,χ),

where the sub-index on the right hand side indicates that we have removed the Euler factors all primes inthe support of c. In particular if we take the character χ trivial (may not primitive) at the primes dividing cthen we have that the two functions are the same.

We start by stating a result of Shimura [23, Theorem 28.8]. We take an f ∈ Sk(C;Q), where

C = x ∈ D[b−1c,bc]|ax−1≺ c

We moreover take f of trivial Nebentypus and assume that it is an eigenform for all Hecke operators awayfrom the primes in the support of c. In the notation of Shimura in Chapter V of his book, we take e= c, andnot e= g. In particular here we take the Euler factors Zv trivial for v in the support of c. The theorem belowis stated only for k ∈ Za.

Theorem 10 (Shimura). With notation as above define m0 := minkv|v ∈ a and assume m0 > (3n/2)+1.Let χ be a character of F such that χa(x) = xt

a|xa|−t with t ∈ Za. Set µv := 0 if kv− tv ∈ 2Z and µv = 1 ifkv− tv 6∈ 2Z. Let σ0 ∈ Z such that

1. 2n+1− kv +µv ≤ σ0 ≤ kv−µv,2. σ0− kv +µv ∈ 2Z for every v ∈ a if σ0 > n,3. σ0−1+ kv−µv ∈ 2Z for every v ∈ a if σ0 ≤ n.

We exclude the cases

1. σ0 = n+1, F =Q and χ2 = 1,2. σ0 = 0, c= g and χ = 1,3. 0 < σ0 ≤ n, c= g, χ2 = 1 and the conductor of χ is g.

Then we haveZ(σ0, f,χ)< f, f >

∈ πn(∑v kv)+deQ

where d = [F : Q] and

e :=(n+1)σ0−n2−n, σ0 > n;nσ0−n2, otherwise.

We now take f ∈ Sk(C,ψ;Q) with C of the form D[b−1,bc] (i.e. the standard setting in this paper). Weare interested in special values of Z′(s, f,χ) for a Hecke character χ of F of conductor f.

Theorem 11. Let f ∈ Sk(b,c,ψ;Q) be an eigenform for all Hecke operators. Assume that either

1. there exists v,v′ ∈ a such that kv 6= kv′ , and m0 = minkv|v ∈ a> [3n/2+1]+2 or2. k is a parallel weight with k > 2n+1.

Let χ be a character of F such that χa(x) = xta|xa|−t with t ∈ Za. Define t ′ ∈ Za by (ψχ)a(x) = xt ′

a |xa|t′. Set

µv := 0 if kv− t ′v ∈ 2Z and µv = 1 if kv− t ′v 6∈ 2Z. Let σ0 ∈ Z such that

1. 2n+1− kv +µv ≤ σ0 ≤ kv−µv for all v ∈ a,2. |σ0−n− 1

2 |+n+ 12 − k+µ ∈ 2Za.

3. if n is even, and σ0 = n/2+ i for i = 0, . . .n/2, i ∈ N or if n is odd and σ0 = n/2−1+ i, i = 1, . . . ,(n+1)/2, then we assume the Assumption below.

We exclude the cases


1. σ0 = n+1, F =Q and (χψ)2 = 1,2. σ0 =

n2 , c= g, n is even and there is no (τ,r) that satisfy our assumption such that ρτ 6= 1 and χψ = 1,

3. n/2 < σ0 ≤ n, c= g and (ψχ)2 = 1.

Then with notation as in the previous theorem we have

Z′(σ0, f,χ)< f, f >

∈ πn(∑v kv)+deQ

Moreover, if we take a number field W so that f, fρ ∈ Sk(W ) and Φ ⊂W, where Φ is the Galois closureof F in Q, then

Z′(σ0, f,χ)

πβ (√

DFn(n+1)/4

)imω(εχψ)ρ < f, f >∈W :=W (χψ),

where ω(·) is defined by using the Theorem 3 as follows

1. for σ0 > n and n even then ω(·) is as in Theorem 3 (i),2. for σ0 > n and n odd then ω(·) is as in Theorem 3 (iii) (b),3. for σ0 ≤ n and n even then ω(·) is as in Theorem 3 (ii),4. for σ0 ≤ n and n odd then ω(·) is as in Theorem 3 (iv).

and m = d if [n/2] is odd and 0 otherwise.

Assumption: Let θ ∈ F×h so that θg = b−1d. Write f′ for the conductor of χ2. We assume that we canfind τ ∈ S+∩GLn(F) and r ∈ GLn(F)h so that c(τ,r; f) 6= 0, equation 11 in Theorem 6 holds and

1. if n is even and v - cf′ then (θ trτr)v is regular and v - f,2. if n is odd and v - cf′ then (θ trτr)v is regular and v - 2f∩b−1d.

We note that this assumption implies that in Theorem 6 we have that Λc(s)/Λh(s) = 1 (see [20, Proposi-tion 8.3]).

Proof. (of Theorem 11) We first consider the Gamma factors that appear in Theorem 6. We first recall that

Γn(s) = πn(n−1)/4

n−1

∏j=0

Γ (s− j2).

Hence for ∏v∈a Γn(σ0−n−1+kv+µv

2 ) we need the condition that σ0 > 2n− kv + µv for all v ∈ a, which isthe lower bound appearing in the theorem. Moreover the Eisenstein series D( ν

2 ) of weight k− µ − n2 for

ν = σ0− n2 is nearly holomorphic if and only if n+ 1− (kv− µv− n

2 ) ≤ σ0− n2 ≤ kv− µv− n

2 and |ν −n+1

2 |+n+1

2 − kv +µv +n2 ∈ 2Z for every v ∈ a. These inequalities give the upper bound in the (i) condition

for σ0 and (ii). The third condition for σ0 is imposed so that in the range where the fraction Λc(s)/Λh(s)(a finite product of Euler factors associated to finite order characters) could have a pole it is equal to 1.Finally the various exclusion follows from various cases where the Eisenstein series D( ν

2 ) is not nearlyholomorphic.

We take β ∈ N so that π−β D( v2 ) ∈ Nk−l(ΦQab). Now using Theorem 6 after a proper choice of (τ,r)

we have

πγ Z′(σ0, f,χ)ψc(det(r))c(τ,r; f) =

α

(D−1/2

F

)n(n+1)/2det(τ)s′0u+λ |det(r)|n+1−σ0

A ×

∏v∈b

gv((ψ/ψc)(πv)χ∗(πvg)|πv|σ0)(Λc/Λh)((2σ0−n)/4)< f ,θχ(π

−β D(ν/2))>,

where α ∈Q×, and γ := n|| k−l−νa2 ||−n||k||+dε−n||s′0u+λ ||−β where we recall ε = n2/4 if n even and

(n2−1)/4 otherwise.We now note that θχ ∈Ml(W) and π−β D(ν/2)∈Nr

k−l(WQab) where r =(k− l−νa)/2 if ν > (n+1)/2and r = (k− l− (n+1−ν)a)/2 otherwise.

Moreover we have s′0u+λ = σ0u+k+µ

2 − n+12 u. In particular


1. for σ0 > n and n even we have that s′0u+λ 6∈ 2Z,2. for σ0 > n and n odd we have that s′0u+λ ∈ 2Z,3. for σ0 ≤ n and n even we have that s′0u+λ ∈ 2Z,4. for σ0 ≤ n and n odd we have that s′0u+λ 6∈ 2Z.

We now note that g(ρτ) = im√

NF/Qdet(τ), with m = d if [n/2] is odd and 0 otherwise. Now we set

P :=√

DFn(n+1)/4imω(εχψ) where ω(·) is defined as in the statement of the theorem. Then by Theorem 3

we conclude that (D−1/2

F

)n(n+1)/2det(τ)s′0u+λ

π−β P−1D(ν/2) ∈Nr

k−l(W).

We set a :=(

D−1/2F

)n(n+1)/2det(τ)s′0u+λ π−β P−1. By Lemma 15.8 in [23] we have that there exists a q ∈

Mk(W) so that < f ,θχ aD(ν/2) >=< f ,q >. If k is not a parallel weight, then we have that actuallyq ∈ Sk(W) since in this case Mk = Sk. Then by Theorem 7 we have that < f ,q>

< f , f> ∈W. In the other case,that is of k being a parallel weight we can use Theorem 8 combined with the Theorem 9 to conclude again< f ,q>< f , f> ∈W and hence conclude the proof. ut

We now obtain also some results with reciprocity laws.

Theorem 12. Let f ∈ Sk(b,c,ψ;Q) be an eigenform for all Hecke operators. With notation as before wetake m0 > [3n/2+1]+2. Let χ be a character of F such that χa(x) = xt

a|xa|−t with t ∈ Za. Define t ′ ∈ Za

by (ψχ)a(x) = xt ′a |xa|t

′. Set µv := 0 if kv− t ′v ∈ 2Z and µv = 1 if kv− t ′v 6∈ 2Z. Assume that µ 6= 0.

Let σ0 ∈ Z be as in the previous Theorem. Then with Ωf ∈ C× as defined in the previous section inTheorem 7 we have for all σ ∈ Gal(Q/Φ) that Z′(σ0, f,χ)

πn(∑v kv)+de√

Dn(n+1)/4F imω(εχψ)ρ Ωf

σ

=

Z′(σ0, fσ ,χσ )

πn(∑v kv)+deim√

DFn(n+1)/4

ω(εχσ ψσ )ρ Ωfσ

.

Proof. We first observe that thanks to the assumption that µ 6= 0 we have that θχ ∈ Sl . Moreover forσ ∈ Gal(Q/Φ) we have θ σ ′

χ = θχσ , as it follows from the explicit Fourier expansion of θχ . Moreoverarguing as in the theorem above and using the reciprocity laws for Eisenstein series in Theorem 3 we havethat (

π−β√

DFn(n+1)/2det(τ)s′0u+λ D(ν/2,εψχρτ)

ω(εψχ)

)σ ′

=

π−β√

DFn(n+1)/2det(τ)s′0u+λ D(ν/2,εψσ χσ )

ω(εψσ χσ ), σ ∈ Gal(Q/Φ).

Moreover we have that θχ D(ν/2,εψσ χσ ρτ)∈Rk. By Proposition 15.6 in [23] we have that there exists q=

p(θχ D(ν/2,εψσ χσ ρτ)

)∈ Sk so that < f ,θχ D(ν/2,εψχρτ)>=< f ,q> and qσ = p

(θ σ

χ D(ν/2,εψχρτ)σ

)for all σ ∈ Aut(C/Φ). In particular we have that(√

DFn(n+1)/2det(τ)s′0u+λ < f ,θχ π−β D(ν/2,εψχρτ)>

ω(εψχ)ρ Ωf

)σ

=

√DF

n(n+1)/2det(τ)s′0u+λ < f σ ,θχσ π−β D(ν/2,εψσ χσ ρτ)>

ω(εψσ χσ )ρ Ωfσ

,

from which we conclude the proof of the theorem. ut

As we have remarked in the introduction results similar to the ones proved in this paper have been ob-tained by Sturm [25], Harris [10] and Panchishkin [17] in the case of F =Q and n even. We also remark that


Sturm has also considered the case n = 1 and F = Q in [26]. Our proofs are just generalizations of theirsbuilding on some new results of Shimura. We close this section by mentioning that the perhaps strongestresult concerning the special values of Siegel modular forms, at least when F = Q and under some othertechnical assumptions, is due to Böcherer and Schmidt [4]. Using the doubling method (see also the nextsection) and some holomorphic differential operators of Böcherer they obtained algebraicity results but as-suming only that the weight of the Siegel modular form is larger than n rather than 3n

2 + 1. It is of coursevery interesting to extend their results to the totally real field case, however the generalization of their resultseems to be a quite challenging task. We comment a bit more on this in the next section.

8 Some Remarks on the Doubling Method

In this paper our main tool for the study of the special L-values of Siegel Modular Forms was the Rankin-Selberg method. However, as we also briefly mentioned above, there is yet another powerful method for thestudy of these special values, namely the so-called doubling method. In this paper we considered mainly theRankin-Selberg Method, since this article, already quite long, would have increased considerably in size ifthe doubling method was also to be considered here. So we have decided to defer the consideration of thedoubling method with respect to the same questions addressed here for a future paper. In this section wewish to very briefly discuss various aspects that are closely related to the doubling method and the questionsconsidered in this paper.

It is perhaps fair to say that the doubling method was initiated by Garrett in [9] and extended further byBöcherer [1, 2, 3], Böcherer and Schmidt [4], Shimura [21] and in the automorphic language by Piatetski-Shapiro and Rallis [PR94]. Of course the list of contributors here is not meant to be complete.

Concerning the algebraicity results addressed in this paper, it seems that the two methods (Rankin-Selberg and the Doubling Method) provide in many cases the same results, but there are indeed case whereone method is better than the other. Indeed, Shimura in his books [23, theorem 28.8] concludes his alge-braicity results by using both methods (1st method and 2nd method in Shimura’s notation). However oneshould at this point remark the following. Shimura writes at the beginning of his proof of his Theorem 28.8“There are two ways to prove this: the first one (i.e. doubling method) applies to the whole critical strip, andthe second one (i.e. Rankin-Selberg Method) only to the right half of the strip”. However this is so, becauseShimura is taking e = c in his book [23, page 231](see also our discussion just before Theorem 7.1 in thispaper). Indeed in this situation the doubling method seems to be able to tackle critical points also to the lefthalf of the critical strip, something that the Rankin-Selberg method cannot. However for e = g this is notthe case and this is the situation that we consider here. The main reason being that the integral expressionin Theorem 6 it is available in this form (in particular this particular Siegel type Eisenstein series for whichwe know quite explicitly) only in the case e= g. We also note here that e= c corresponds to Γ1(N)-case andc= g corresponds to Γ0(N)-case in the elliptic modular form situation.

In this paper we have considered only Siegel modular forms. Of course the same questions can be ad-dressed for other groups, as for example unitary groups. Actually Shimura in his book provides similarresults (always over an algebraic closure of Q) for hermitian modular forms, that is modular forms asso-ciated to unitary groups. For hermitian modular forms the two methods are not at all equivalent, and inparticular one cannot use the Rankin-Selberg method to study special L-values for hermitian forms of uni-tary groups of the form U(n,m) for n 6= m (this is part of the case UB in Shimura’s notation in his book).For example one cannot consider the case of hermitian modular forms for definite unitary groups. But thedoubling method still does apply. Here we should say that the field of definition for hermitian modular formshas been worked out by Harris in [13] using the doubling method. However we would like further to remarkhere that Harris considers special L-values only in the strip of absolute convergence. In the UT case (i.e.n = m) we have obtained in some cases results [6] using the Rankin-Selberg method that improves the onesof Harris (i.e. beyond the absolute convergence).

As we explained at the beginning of this article one of the main motivations of our investigations isthe construction of p-adic measures for Siegel modular forms. We briefly describe here what is knownwith respect to this, even though the reader should keep in mind that we do not wish to give here a com-plete and detailed picture of the situation. Historically the first results in this direction were obtained byPanchishkin,[17] (see also the joint work of Courtieu and Panchishkin [7]), who used the Rankin-Selberg


method to construct these measures. However he considered the case of even degree (or genus), the mainreason being that in this case the Rankin-Selberg method does not involve Eisenstein series of half-integralweight. Later Böcherer and Schmidt [4] constructed these p-adic measures for any degree using the dou-bling method. One should remark here that there is a very delicate difference in the way that Böcherer andSchmidt applied the doubling method and in the way Shimura developed it in his work [21]. Very briefly themain difference seems to be in the decomposition that is proved in Proposition 2.1 of [4] as well as the useof the holomorphic operators of Böcherer (opposite to the non-holomorphic ones in the work of Shimura).Of course one should add here that the work [4] is restricted to Siegel modular forms over the rationals,opposite to the work of Shimura who applies to any totally real field. We simply say here that in an ongoingproject we extend the work of Panchishkin (i.e. p-adic measures using the Rankin-Selberg mathod) in twodirections. We consider also odd genus and to the totally real field case. Note, as we already said, that boththe work of Panchishkin and of Böcherer and Schmidt are over the rationals. It seems to be a big challengeto obtain the analogue of Proposition 2.1 of [4] in the totally real case in the situation of strict class numberbigger than one, and in particular extend the work of Böcherer and Schmidt to totally real fields. We arecurrently working on this. At this point it is worth mentioning that in this article we considered scalar valuedSiegel modular forms. Many of the above questions can be stated also for the vector valued ones. For a firststep in this direction the reader can see [16]. Finally we close this article by mention that of course it isvery interesting to construct p-adic measures for hermitian modular forms. The doubling method has beenalready used in that context, as for example in [14, 15, 24]. In [6] we are considering the Rankin-Selbergmethod for constructing these p-adic measures.

References

1. S. Böcherer, Über die Fourier-jacobi-Entwicklung der Siegelschen Eisensteinreihen, Math. Z. 183 (1983), 21-432. S. Böcherer, Über die Fourier-jacobi-Entwicklung der Siegelschen Eisensteinreihen II, Math. Z. 189 (1985), 81-1003. S. Böcherer, Über die Funktionalgleichung automorpher L-Funktionen zur Siegelschen Modulgruppe. J. R. Ang. Math.

362, 146 168 (1985)4. S. Böcherer and C.-G. Schmidt, p-adic measures attached to Siegel modular forms, Annales de l’ institut Fourier, tome

50, no. 5 (2000), p. 1375-14435. Th. Bouganis, On special L-values attached to metaplectic modular forms, in preparation6. Th. Bouganis, p-adic measures for hermitian modular forms and the Rankin-Selberg method, in preparation7. M. Courtieu and A.A. Panchishkin, , Non-Archimedean L-Functions and Arithmetical Siegel Modular Forms, Lecture

Notes in Mathematics 1471, Springer-Verlag, 2004 (2nd augmented ed)8. P. Feit, Poles and residues of Eisenstein series for symplectic and unitary groups, Memoirs of the American Mathematical

Society, Number 346, vol 61, 19869. P. B. Garrett, Pullbacks of Eisenstein Series; Applications, In Satake I, Morita Y (eds) Automorphic Forms in Several

Variables,pp. 114-137. Boston: Birkhäuser, (1984)10. P. B. Garrett, On the arithmetic of Siegel-Hilbert cuspforms: Petersson inner products and Fourier coefficients, Invent.

math. 107, 453-481, (1992)11. M. Harris, Special values of zeta functions attached to Siegel modular forms, Ann. Sci. Éc Norm. Super. 14, 77-120 (1981)12. M. Harris, Eisenstein Series on Shimura varieties, Ann. Math. 119, 59-94 (1984)13. M. Harris, L functions and periods of polarized regular motives, Journal für die reine und angewandte Mathematik, 483,

1997 (75-161)14. M. Harris, J.-S. Li, and C. Skinner, The Rallis inner product formula and p-adic L-functions, Automorphic Representations,

L-Functions and Applications: Progress and Prospects, Volume in honor of S.Rallis, Berlin:de Gruyter, 225-255 (2005)15. M. Harris, J.-S. Li, and C. Skinner, p-adic L-functions for unitary Shimura varieties, I: Construction of the Eisenstein

measure, Documenta Math., Extra Volume: John H.Coates’ Sixtieth Birthday 393-464 (2006)16. T. Ichikawa, Vector-valued p-adic Siegel modular forms, J. reine angew.Math., DOI 10.1515/crelle-2012-006617. A.A. Panchishkin, Non-Archimedean L-functions of Siegel and Hilbert Modular Forms, Lecture Notes in Math. 1471,

Springer-Verlag (1991) 166 p.18. I. Piatetski-Shapiro and S. Rallis, L-functions for classical groups, Lecture notes, Institute for Advanced Study, 198419. G. Shimura, Confluent hypergeometric functions on tube domains, Math. Ann. 260 (1982), 269-30220. G. Shimura, Euler Products and Fourier Coefficients of automorphic forms on symplectic groups, Invent. math. 116,

531-576 (1994)21. G. Shimura, Eisenstein series and zeta functions on symplectic groups, Inventiones Mathematicae 119 , 539-584, (1995)22. G. Shimura, Euler Products and Eisenstein Series, CBMS Regional Conference Series in Mathematics, No. 93, Amer.

Math. Soc. 199723. G. Shimura, Arithmeticity in the Theory of Automorphic Forms, Mathematical Surveys and Monographs, Vol 82, Amer.

Math. Soc. 200024. C. Skinner, E. Urban, The Iwasawa Main Conjectures for GL2, Invent. math DOI 10.1007/s00222-013-0448-1


25. J. Sturm, The critical values of zeta functions associated to the symplectic group, Duke Mathematical Journal vol 48, no2 (1981)

26. J. Sturm, Special Values of Zeta Functions, and Eisenstein Series of Half Integral Weight, American Journal of Mathe-matics, Vol. 102, No. 2 (Apr., 1980), pp. 219-240

Modular symbols in Iwasawa theory

Takako Fukaya, Kazuya Kato, and Romyar Sharifi

Abstract This survey paper is focused on a connection between the geometry of Gld and the arithmetic ofGld−1 over global fields, for integers d ≥ 2. For d = 2 over Q, there is an explicit conjecture of the thirdauthor relating the geometry of modular curves and the arithmetic of cyclotomic fields, and it is proven inmany instances by the work of the first two authors. The paper is divided into three parts: in the first, weexplain the conjecture of the third author and the main result of the first two authors on it. In the second, weexplain an analogous conjecture and result for d = 2 over Fq(t). In the third, we pose questions for generald over the rationals, imaginary quadratic fields, and global function fields.

1 Introduction

1.0.1. The starting point of this paper is the fascinatingly simple and explicit map

[u : v] 7→ 1−ζuN ,1−ζ

vN

that relates the worlds of geometry/topology and arithmetic [Bu, Sh]. Here,

• [u : v] is a Manin symbol in the relative homology group H1(X1(N),cusps,Z), and• 1−ζ u

N ,1−ζ vN is a Steinberg symbol in the algebraic K-group K2(Z[ζN ,

1N ]),

for N ≥ 1, where u,v∈Z/NZ are nonzero numbers with (u,v) = (1), and ζN is a primitive Nth root of unity.

1.0.2. The above map connects two different worlds in the following manner:

geometric theory of Gl2 =⇒ arithmetic theory of Gl1

over the field Q. Here, if we consider the geometry of the modular curve X1(N) on the left, then we considerthe arithmetic of the cyclotomic field Q(ζN) on the right. This connection is conjectured to be a correspon-dence if we work modulo the Eisenstein ideal that is defined in 2.1.6:

geometric theory of Gl2 modulo the Eisenstein ideal⇐⇒ arithmetic theory of Gl1 .

More generally, we are dreaming that there is a strong relationship

Takako FukayaDepartment of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, Illinois 60637, USA, e-mail: [email protected]

Kazuya KatoDepartment of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, Illinois 60637, USA, e-mail: [email protected]

Romyar SharifiDepartment of Mathematics, University of Arizona, 617 N. Santa Rita Ave., Tucson, Arizona 85711, USA, e-mail: [email protected]

133

[email protected]

[email protected]

[email protected]

[email protected]

[email protected]

[email protected]

134 Takako Fukaya, Kazuya Kato, and Romyar Sharifi

geometric theory of Gld modulo the Eisenstein ideal⇐⇒ arithmetic theory of Gld−1

over global fields. Our goals are to survey what is known and to explain this dream.

1.0.3. The connection with the Eisenstein ideal for Gl2 over Q appears as follows. The homology group weconsider has the action of a Hecke algebra which contains an Eisenstein ideal, and the map of 1.0.1 factorsthrough the quotient of the homology by this ideal [FK]. The truth of this is deep and mysterious; it is theidea of specializing at the cusp at ∞. This is the key to the connection between Gl2 and Gl1.

1.0.4. We note that there exist two technical issues with our simple presentation of the “map” in 1.0.1. Weleft out those Manin symbols in which one of u or v is 0, which are needed to generate the relative homologygroup. Also, the map is only well-defined as stated if we first invert 2 and then project to the fixed part undercomplex conjugation (see Section 2.1).

1.0.5. Let us consider the case that N is a power of an odd prime p and work only with p-parts. Considerthe quotient

Pr = H1(X1(pr),Zp)+/IrH1(X1(pr),Zp)

+

of the fixed part of homology under complex conjugation by the action of the Eisenstein ideal Ir in thecuspidal Hecke algebra of weight 2 and level pr. By the well-known relationship between K2 and H2

ét ofZ[ζpr , 1

p ], the map of 1.0.1 yields a well-defined map

ϖr : Pr→ H2ét(Z[ζpr , 1

p ],Zp(2))+

that sends the image of [u : v] in Pr to the cup product (1−ζ upr)∪ (1−ζ v

pr).

1.0.6. Let us connect this with Iwasawa theory for Gl1. As we increase r, the maps ϖr are compatible. Thegroup H2

ét(Z[ζpr , 1p ],Zp(2))+ is related to the p-part Ar of the class group of Q(ζpr) in the sense that its

reduction modulo pr is isomorphic to the Tate twist of A−r /prA−r . So, if we let P = lim←−rPr and X = lim←−r

Ar,then we obtain a map

ϖ : P→ X−(1)

that relates geometry of the tower of curves X1(pr) modulo the Eisenstein ideal to Iwasawa theory over theunion of cyclotomic fields Q(ζpr). It is a map of Iwasawa modules under the action of inverses of diamondoperators on the left and of Galois elements on the right.

1.0.7. In [Sh], the map ϖ is conjectured to be an isomorphism. If this conjecture is true, we can understandthe arithmetic object X− by using the geometric object P. The Iwasawa main conjecture states that thecharacteristic ideal of X− is the equivariant p-adic L-function. On the other hand, the characteristic ideal ofP under the inverse diamond action can be computed to be a multiple of the Tate twist ξ of this L-function. Ifthe characteristic ideals of X−(1) and P are equal, then the main conjecture follows as a consequence of theanalytic class number formula. Therefore, the conjecture that ϖ is an isomorphism is an explicit refinementof the Iwasawa main conjecture.

1.0.8. In their proof of Iwasawa main conjecture [11], Mazur and Wiles, expanding upon the work of Ribet[Ri], considered the relationship between the geometric theory of Gl2 and the arithmetic theory of Gl1.Using roughly their methods, we can define a map

ϒ : X−(1)→ P.

More precisely, ϒ is constructed out of the Galois action on the projective limit of the reduction of étalehomology groups Hét

1 (X1(pr)/Q,Zp) modulo the Eisenstein ideal. The expectation in [Sh] is that the mapsϖ : P→ X−(1) and ϒ : X−(1)→ P are inverse to each other. The best evidence we have for this is theequality ξ ′ϒ ϖ = ξ ′ after multiplication by the derivative ξ ′ of ξ , which is proven in [FK]. If the p-adicL-function ξ has no multiple zeros, this yields the conjecture up to p-torsion in P.

1.0.9. An analogous result for the rational function field Fq(t) over a finite field can be proven by followingthe analogy between Fq(t) and Q. In both cases, the key point of the proof is that (1− ζ u

N ,1− ζ vN) and its

analogue for Fq(t) are values at the infinity cusp of the “zeta elements,” which is to say Beilinson elementsand their analogues for Fq(t), which live in K2 of the modular curve X1(N) and its Drinfeld analogue forFq(t).

Modular symbols in Iwasawa theory 135

1.0.10. For both Q and Fq(t), the philosophy is that ξ ′ϒ ϖ is the reduction modulo the Eisenstein idealI of a map involving zeta elements. Roughly speaking, the proof consists firstly of the demonstration ofthe existence of a commutative diagram Here, S is the space of modular symbols, the map z takes modularsymbols to zeta elements in the K2-group K of a modular curve, S is a space of p-adic cusp forms, reg is thep-adic regulator map, and Y is either X−(1) or its analogue for Fq(t). The vertical arrows denoted by “modI” are obtained by reduction modulo the Eisenstein ideal I (see Section 2.7 for details), and ∞ is given byspecialization at a cusp at infinity. Secondly, it entails a computation of the regulator map on zeta elementsthat tells us that the composition S→ K→S→ P coincides with ξ ′ times the projection S→ P.

1.0.11. In this survey paper, we explain the key ideas and concepts of our work, putting aside many of thetechnical details that must arise in a careful treatment. While we do our best to strike a balance, the readershould be aware that some of the statements we make require minor modifications in order that they hold.

The structure of the paper is as follows. In Section 2, we describe the original case of the conjecturesand outline the proof of the above result. In Section 3, we discuss and outline the proof of the analogue forFq(t). In Section 4, we describe what might be expected for Gld .

2 The case of Gl2 over Q

Fix an odd prime number p and an integer N ≥ 1 which is not divisible by p. Let r≥ 1, which will vary. LetQ be the algebraic closure of Q in C, and let us fix an embedding of Q in Qp.

Recall that we want to understand the picture: In Sections 2.1-2.3, we study the map ϖ . In Sections 2.4and 2.5, we study the map ϒ . In Sections 2.6 and 2.7, we state the conjecture and the main result on it.

2.1 From modular symbols to cup products

We construct the map ϖr that relates modular symbols in the homology of X1(N pr) to cup products in thecohomology of the maximal unramified outside N p extension of Q(ζN pr).

2.1.1. We introduce homology groups Sr and Mr of modular curves.Let H denote the complex upper half-plane and Γ1(N pr) the usual congruence subgroup of matri-

ces in SL2(Z) that are upper-triangular and unipotent modulo N pr. We consider the complex pointsY1(N pr) = Γ1(N pr)\H of the open modular curve over C. It is traditional to use cusps to denote thecusps Γ1(N pr)\P1(Q), but let us instead set Cr = cusps. We let

X1(N pr) = Y1(N pr)∪Cr = Γ1(N pr)\H∗,

be the closed modular curve, where H∗ =H∪P1(Q) is the extended upper half-plane.The usual modular symbols lie in the first homology group of the space X1(N pr) relative to the cusps.

However, H1(X1(N pr),Cr,Z) is not exactly the natural object for our study. Rather, we are interested inits quotient by the action of complex conjugation, the plus quotient.1 We consider the plus quotients ofhomology and homology relative to the cusps:

Sr = H1(X1(N pr),Z)+ and Mr = H1(X1(N pr),Cr,Z)+,

where ( )+ denotes the plus quotient.

2.1.2. We introduce Manin symbols [u : v]r ∈Mr.Let u,v ∈ Z/N prZ be such that (u,v) = (1). For such u and v, we can find γ =

(a bc d

)∈ SL2(Z) with

u = c mod N pr and v = d mod N pr. Define

[u : v]r =

dbN pr →

caN pr

r,

1 This is still not quite the right object unless we invert 2. In Section 4, we take the point of view that the right object is therelative homology of the quotient of the space X1(N pr) by the action of complex conjugation.


where α → βr for α,β ∈ P1(Q) denotes the class in Mr of the hyperbolic geodesic on H∗ from α to β .Then [u : v]r is independent of the choice of γ .

By the work of Manin [Ma], we have that the group Mr of modular symbols is explicitly presented as anabelian group by generators [u : v]r and relations

[u : v]r = [−u : v]r =−[v : u]r and [u : v]r = [u : u+ v]r +[u+ v : v]r.

2.1.3. We define an intermediate relative homology group M0r used in constructing ϖr.

We do not use all modular symbols to connect with Gl1. Rather, we use those modular symbols withboundaries in cusps that do not lie over the cusp at 0 in X0(N pr) = Γ0(N pr)\H∗. Let us denote the set ofcusps of X1(N pr) that do not lie over the 0-cusp of X0(N pr) by C0

r . The intermediate space

M0r = H1(X1(N pr),C0

r ,Z)+

is the largest space on which we may define ϖr and have it factor through the Eisenstein quotient (see 2.1.7).We have Sr ⊂M0

r ⊂Mr.Our intermediate space also has a simple presentation: it is generated by the [u : v] for nonzero u,v ∈

Z/N prZ with (u,v) = (1), together with the relations of 2.1.2, again for nonzero u and v, and excluding thelast relation when u+ v = 0.

2.1.4. We define the map ϖr, which gives our first connection between Gl2 and Gl1.We start with the primitive N prth root of unity ζN pr = e2πi/N pr

. It generates the cyclotomic field Er =Q(ζN pr) and its integer ring Z[ζN pr ]. Inside Er, we have the maximal totally real subfield Fr = Q(ζN pr)+

and its integer ring Or = Z[ζN pr ]+.For a,b∈Z[ζN pr , 1

N p ]×, we let a,br denote the norm of the Steinberg symbol of a and b to K2(Or[

1N p ]).

There is a well-defined homomorphism

ϖr : M0r ⊗Z[ 1

2 ]→ K2(Or[1

N p ])⊗Z[ 12 ], [u : v]r 7→ 1−ζ

uN pr ,1−ζ

vN prr

for u,v 6= 0. Using the Steinberg relation x,1− xr = 0 in K2 for x,1− x ∈ Z[ζN pr , 1N p ]×, one may easily

check that the 1−ζ uN pr ,1−ζ v

N prr satisfy the same relations as the [u : v]r (see [Sh, Bu] for instance). It isnecessary to invert 2 for these relations to hold.

2.1.5. We interpret ϖr on p-completions in terms of cup products in Galois cohomology.For a commutative ring R in which p is invertible, the Kummer exact sequence

0→ Z/pnZ(1)→Gmpn

→Gm→ 0

on Spec(R)ét induces the connecting map R× → H1ét(R,Z/pnZ(1)). We have also the Chern class map

K2(R)→ H2ét(R,Z/pnZ(2)). The value of this map on a product (i.e., Steinberg symbol) in K2(R) of a pair

of elements of R× is equal to the cup product of the images in H1ét(R,Z/pnZ(1)) of the two elements.

We may apply this discussion with R equal to Z[ζN pr , 1N p ] or Or[

1N p ], in which cases the Chern class

map K2(R)⊗Zp → H2ét(R,Zp(2)) is an isomorphism [26]. Moreover, the diagram commutes, where N is

induced by the norm and cor is induced by corestriction. The map cor is an isomorphism as Or[1

N p ] hasp-cohomological dimension 2. Let (1− ζ u

N pr ,1− ζ vN pr)r denote the corestriction of the cup product of the

elements 1−ζ uN pr and 1−ζ v

N pr of

Z[ζN pr , 1N p ]×⊗Zp

∼−→ H1ét(Z[ζN pr , 1

N p ],Zp(1)).

By definition of the symbols, the Chern class map in the lower row of the diagram satisfies

1−ζuN pr ,1−ζ

vN prr 7→ (1−ζ

uN pr ,1−ζ

vN pr)r.

We will study the homomorphism to Galois cohomology

ϖr : M0r ⊗Zp→ H2

ét(Or[1

N p ],Zp(2)), [u : v]r 7→ (1−ζuN pr ,1−ζ

vN pr)r,

which is identified with our original ϖr on p-completions.


2.1.6. We define Hecke algebras Tr and Tr and their Eisenstein ideals Ir and Ir.The Hecke operators T (n) for n ≥ 1 generate a subalgebra Tr of EndZp(Mr⊗Zp), the modular Hecke

algebra. We will be interested in this section only in its action on M0r . We also have a cuspidal Hecke algebra

Tr of EndZp(Sr⊗Zp) and a canonical surjection TrTr. These Hecke algebras contain diamond operators〈d〉 for d ∈ Z, which we take to be 0 if (d,N p) 6= 1.

The Hecke algebra Tr contains the Eisenstein ideal Ir generated by the T (n)−∑d|n d〈d〉 for n≥ 1. It isalso generated by T (`)−1− `〈`〉 for primes `. The image Ir of Ir in Tr is an Eisenstein ideal with the samegenerators.

2.1.7. We connect our study of ϖr with the Eisenstein ideal.The third author conjectured [Sh] (on Sr, see also [Bu] for N = 1), and the first two authors proved [FK,

Theorem 5.2.3] that ϖr is “Eisenstein.” By this, we mean that ϖr factors through the quotient of M0r ⊗Zp

by the Eisenstein ideal, that is, through a map

(M0r/IrM

0r )⊗Zp→ H2

ét(Or[1

N p ],Zp(2)).

We can show that this follows from the fact that ϖr is the specialization of a map in the Gl2-setting: seeSection 2.3.

2.1.8. Let Gr = (Z/N prZ)×/±1, and set Λr = Zp[Gr]. The algebra Λr appears in two different contextsin our story:

(2)

(1) On the Gl2-side, Λr is a Zp-algebra of diamond operators in Tr (or Tr): we define a Zp-linear injectionιr : Λr → Tr that sends the group element in Λr corresponding to a ∈ (Z/N prZ)×/〈−1〉 to the inverse〈a〉−1 of the diamond operator for a (i.e., for any lift of a to an integer).

(2) On the Gl1-side, Λr is the Zp-group ring of Gal(Fr/Q): we have an isomorphism

(Z/N prZ)× ∼−→ Gal(Er/Q), a 7→ σa,

where σa(ζN pr) = ζ aN pr . This gives rise to an isomorphism Gr

∼−→ Gal(Fr/Q) that is the map on groupelements defining Λr

∼−→ Zp[Gal(Fr/Q)].

These actions are compatible with ϖr in the sense that for any x ∈M0r ⊗Zp and a ∈ (Z/N prZ)×, we

haveϖr(〈a〉−1x) = σaϖr(x).

This is easily seen: taking x = [u : v]r for some nonzero u and v, we have

〈a〉−1[u : v]r = [au : av]r and σa(1−ζuN pr ,1−ζ

vN pr)r = (1−ζ

auN pr ,1−ζ

avN pr)r.

So, to say that ϖr is Eisenstein is to say that ϖr(T`x) = (1+ `σ−1` )ϖr(x) for primes ` - N p and ϖr(T`x) =

ϖr(x) for ` | N p.

2.2 Passing up the modular and cyclotomic towers: the map ϖ

We pass up the modular tower on the Gl2-side and the cyclotomic tower on the Gl1-side to define the mapϖ = lim←−r

ϖr.

2.2.1. Let G = lim←−rGr. Then the completed group ring

Λ = ZpJGK = lim←−r

Λr

is the Iwasawa algebra for G. As with Λr, let us emphasize its dual nature: (2)

(1) Set T = lim←−rTr and T = lim←−r

Tr. The projective limit of the injections ιr defines a map ι : Λ → T ofprofinite Zp-modules that takes a ∈ G to the projective system of inverses 〈a〉−1 of diamond operatorscorresponding to a.


(2) Set K = ∪r≥1Fr, the maximal totally real subfield of L = ∪r≥1Er. Then our identifications Λr∼−→

Zp[Gal(Fr/Q)] for r ≥ 1 induce an isomorphism Λ∼−→ ZpJGal(K/Q)K of completed group rings in

the projective limit.

2.2.2. We have the following projective limits of spaces of modular symbols:

S= lim←−r(Sr⊗Zp) and M0 = lim←−

r(M0

r ⊗Zp).

Let I⊂ T and I ⊂ T be the Eisenstein ideals, defined by the same set of generators as Ir, but now viewed ascompatible sequences of operators in the Hecke algebras.

Our maps ϖr are compatible with change of r and induce in the projective limit a map

ϖ : M0→ lim←−r

H2ét(Or[

1N p ],Zp(2))

that factors through M0/IM0 by the result of [FK]. This map ϖ is a homomorphism of Λ -modules, theactions arising from part (1) of 2.2.1 on the left and part (2) of 2.2.1 on the right.

2.2.3. We recall the unramified Iwasawa module X , study its difference from Galois cohomology, andconsider a related Λ -module Y .

Let X be the projective limit of the p-parts Ar of the ideal class groups of the fields Er. Class field theoryallows us to identify X with the Galois group of the maximal unramified abelian pro-p extension of L.

For R as in 2.1.5, the Kummer exact sequence induces

Pic(R) = H1ét(R,Gm)→ H2

ét(R,Z/pnZ(1)).

Taking a projective limit of such maps for the rings R = Or[1

N p ], we obtain

X = lim←−r

Ar→ lim←−r

H2ét(Or[

1N p ],Zp(1)).

In general, this map is neither injective nor surjective. Its kernel and cokernel can be explicitly described ascontributions of classes of primes and Brauer groups at places dividing N p, respectively. We will deal witha part of cohomology on which this subtle difference disappears.

The Iwasawa algebra ZpJGal(L/Q)K acts on X , but this action does not in general factor through Λ . Wewant to consider the (−1)-eigenspace X− of X under complex conjugation. To do so, we take the Tate twistY = X−(1), or equivalently, the fixed part X(1)+. Then σ−1 acts trivially on Y , so Y is a Λ -module. Themap from X to cohomology induces a Λ -module homomorphism

Y → lim←−r

H2ét(Or[

1N p ],Zp(2)).

Together with ϖ , this will allow us to relate S with Y .

2.2.4. We have two objects of study:

• the geometric object P = S/IS for Gl2,• the Iwasawa-theoretic object Y = X−(1) for Gl1.

We can relate these on θ -parts for suitable even characters θ of (Z/N pZ)×.For a primitive, even character θ : (Z/N pZ)×→Qp

×, we may consider the quotient Λθ =Λ⊗Zp[(Z/N pZ)×]Zp[θ ] of Λ , where Zp[θ ] is the Zp-algebra generated by the values of θ . For a Λ -module M, we then letMθ = M⊗Λ Λθ denote its θ -part.

We need a technical assumption to insure that the maps

Pθ →M0θ/IθM

0θ and Yθ → lim←−

rH2

ét(Or[1

N p ],Zp(2))θ

are isomorphisms. Together with primitivity, the assumption is as follows:

• θ |(Z/pZ)× 6= ω or θω−1|(Z/NZ)×(p) 6= 1,


where ω : (Z/N pZ)× → Z×p denotes the Teichmüller character (i.e., projection to (Z/pZ)× ⊂ Z×p ). Forsuch a θ , our ϖ induces a map on θ -parts ϖ : Sθ → Yθ that will factor through Pθ .

2.3 Zeta elements: ϖ is “Eisenstein”

We sketch the proof that ϖ factors through the quotient of M0 by the Eisenstein ideal I.

2.3.1. Let Y1(N pr) be the moduli space of pairs (E,e) where E is an elliptic curve and e is a point of orderN pr on E, and let Y (N pr) be the moduli space of elliptic curves endowed with a full N pr-level structure.We view these moduli spaces as schemes over Z[ 1

N p ]. For any nonzero (α,β ) ∈ 1N pr Z2/Z2, there is a Siegel

unit gα,β ∈ O(Y (N pr))×.2 It has the q-expansion

gα,β = q1

12−α2 +

α22

∞

∏n=0

(1−qn+α e2πiβ )∞

∏n=1

(1−qn−α e−2πiβ ) ∈ Or[1

N p ]Jq1/12N prK[q−1]×.

If α = 0, then we may view g0,β as an element of O(Y1(N pr))×. The crucial point is that the specializationof a Siegel unit of the form g0, u

N pr at the ∞-cusp is the cyclotomic N p-unit 1− ζ uN pr . Specifically, this

specialization is given by projecting its q-expansion to Or[1

N p ]Jq1/N prK× and then evaluating at q = 0 [FK,

Section 5.1].

2.3.2. We have a homomorphism

zr : M0r → H2

ét(Y1(N pr),Zp(2)), zr([u : v]r) = g0, uN pr ∪g0, v

N pr

of Tr-modules that takes a Manin symbol to a Beilinson element given by a cup product of two Siegel units[FK, Proposition 3.3.15]. Related elements were studied in [Ka].

2.3.3. There is again a specialization-at-∞ map

∞r : H2ét(Y1(N pr),Zp(2))→ H2

ét(Or[1

N p ],Zp(2))

that takes g0, uN pr ∪ g0, v

N pr to (1− ζ uN pr ,1− ζ v

N pr)r. So, cup products of cyclotomic units are specializationsat cusps of Beilinson elements. We have

ϖr = ∞r zr : M0r → H2

ét(Or[1

N p ],Zp(2)).

It can be shown that specialization at ∞ is Eisenstein. Hence so is ϖr [FK, Sections 5.1-5.2].

2.3.4. By passing the projective limit over r, we see that ϖ is Eisenstein. The identity ϖ = ∞ z is thecommutativity of the left-hand square in the diagram of 1.0.10.

2.4 Ordinary homology groups of modular curves

Homology groups of the modular curves are useful for us in two different ways. They contain modularsymbols, allowing us to define ϖ . They also have Galois actions, allowing us to define ϒ , which is our nextgoal. We use two different groups derived from homology, S as above and T defined below, to constructthe two maps. For the modular symbols, we require only the plus part of homology. On the other hand, tohave Galois actions, we cannot restrict to plus parts. Instead, we take ordinary parts to control the growth ofhomology groups in the modular tower and to specify the form of the local Galois action at p. The fact thatwe use different groups should be kept in mind in the Gld-setting, in which we will not consider ϒ .

2 Actually, gα,β is a root of a unit, but the difficulties this causes are resolvable by passing to the projective limit and descending,so we ignore this for simplicity of presentation. We will be very careless about denominators in several places, omitting themwhere they occur for simplicity of the discussion that follows.


2.4.1. We introduce Hida’s ordinary p-adic cuspidal and modular Hecke algebras h and H.Recall our cuspidal Hecke algebra T from 2.2.1, which acts Λ -linearly on S. The action of T (p) breaks

it into a direct product of two rings: an ordinary part in which the image of T (p) is invertible and anotherpart in which T (p) is topologically nilpotent. The ordinary cuspidal p-adic Hecke algebra h= Tord of Hida[Hi] is this ordinary part. This is a Λ -subalgebra that is projective of finite Λ -rank. We may speak of Heckeoperators T (n) ∈ h by taking the images of the T (n) ∈ T.

The Hecke algebra h is remarkable in that it simply encapsulates information about the ordinary Heckealgebras of all weights ≥ 2 and all levels dividing some N pr. For instance, its quotient for the action of thekernel of G→Gr is the Hecke algebra hr = Tord

r . This highly regular behavior is the subject of Hida theory.We also have the ordinary modular Hecke algebra H = Tord, of which h is a quotient. In general, if M

is a T-module (resp., T-module), then we use Mord to denote its ordinary part, the maximal summand onwhich T (p) acts invertibly, which is an H-module (resp., h-module).

2.4.2. We introduce the ordinary homology groups T and T. These have commuting actions of Heckealgebras and the absolute Galois group GQ = Gal(Q/Q). The study of these actions on T will allow us todefine the map ϒ in Section 2.5.

The Hecke operators T (n) with n≥ 1 act on the homology of X1(N pr)(C) and the homology relative tothe cusps and are compatible with projective limits. We consider the ordinary parts T and T of the projectivelimits

T = lim←−r

H1(X1(N pr),Zp)ord ⊂ T = lim←−

rH1(X1(N pr),Cr,Zp)

ord.

We are primarily interested in T. The T-action on T factors through h. As an h-module, T is finitely gener-ated and torsion-free, and T is projective of finite rank over Λ . If we denote by Q(Λ) the total quotient ringof Λ , then Q(Λ)⊗Λ T is a free Q(Λ)⊗Λ h-module of rank 2.

The absolute Galois group GQ = Gal(Q/Q) acts on the homology of X1(N pr) by its duality with coho-mology and the identification of Betti cohomology with étale cohomology of the scheme X1(N pr)/Q overQ. This describes the first of the two isomorphisms

H1(X1(N pr),Zp)∼= Hom(H1ét(X1(N pr)/Q,Zp),Zp),

H1(X1(N pr),Cr,Zp)∼= Hom(H1ét,c(Y1(N pr)/Q,Zp),Zp),

where in the second, the duality of the relative cohomology group is with the compactly supported coho-mology of the open modular curve. This Galois action commutes with the action of the Hecke operators, sopasses to ordinary parts, and it is compatible in the towers. Therefore, H[GQ] acts compatibly on T and T.

2.4.3. We introduce the Eisenstein ideals I and I of the ordinary Hecke algebras.Let us reuse the notation I, allowing it to denote the Eisenstein ideal of h, which is the image of the

Eisenstein ideal I of T in h. We remark that, since T (p)−1 ∈ I and 1 is a unit, the quotient map T/I→ h/Iis an isomorphism. We will also reuse the notation I for the Eisenstein ideal of H, the image of I⊂ T.

2.4.4. In the Gl2-setting over Q, there are two places which play important roles: the place at p and the realplace. We study the actions of the corresponding local Galois groups.

We first study the local action at p: here we have an interesting quotient Tquo. The fact that T is ordinaryfor T (p) tells us about the action of GQp , which is to say that it is ordinary in the sense of p-adic Hodgetheory. More specifically to our case, we have an exact sequence

0→ Tsub→ T→ Tquo→ 0

of h[GQp ]-modules, with Tsub and Tquo defined as follows. First, Tsub is the largest submodule of T such thatGQp acts on Tsub(−1) by inverse diamond operators, and Tquo is the quotient. Put more simply, Tquo is themaximal unramified, h-torsion-free quotient of T.

At the real place, we have T+, which is isomorphic to Sord. It fits in an exact sequence

0→ T+→ T→ T/T+→ 0

of h[GR]-modules, and both Q(Λ)⊗Λ T+ and Q(Λ)⊗Λ T/T+ are free of rank 1 over Q(Λ)⊗Λ h.


The compositions T+→ T→ Tquo and Tsub→ T→ T/T+ relate the two exact sequences. We study thesemaps on Eisenstein components in 2.5.5. The interplay between the reductions modulo I of the two exactsequences allows us to construct the map ϒ .

2.4.5. We discuss Λ -adic cusp forms and modular forms and their ordinary parts S and M.Let S2(N pr)Z denote the space of cusp forms of weight 2 and level N pr with integer coefficients. For a

ring R, we then set S2(N pr)R = S2(N pr)Z⊗R. If ε : Gr → R× is a homomorphism, then we may speak ofS2(N pr,ε)R, those cusp forms in S2(N pr)R with nebentypus ε .

Any finite order character ε : G→Qp× induces a ring homomorphism Λ→Qp. We let ε : ΛJqK→QpJqK

be the induced map on coefficients. An element f ∈ΛJqK is said to be a Λ -adic cusp form of weight 2 andlevel N p∞ if for every ε , one has ε( f ) ∈ S2(N pr,ε)Qp

with r ≥ 0 such that ε factors through Gr [?, Oh1].We denote the set of such Λ -adic cusp forms by SΛ .

The Hecke operators T (n) for n ≥ 1 act on SΛ via the usual formal action of Hecke operators on q-expansions. We define S to be the ordinary part Sord

Λof SΛ .3 The ordinary Λ -adic cusp forms and the

ordinary Hecke algebra are dual in the usual sense. That is, we have a perfect pairing of Λ -modules,

h×S→Λ , (T, f ) 7→ a1(T f ),

where a1(g) denotes the q-coefficient in the q-expansion of g ∈ SΛ . As a consequence, Q(Λ)⊗Λ S is freeof rank 1 over Q(Λ)⊗Λ h.

Similarly, we have a space M of ordinary Λ -adic modular forms with q-expansions that are integraloutside of the constant term, which sits inside Q(Λ)+ΛJqK. There is a perfect pairing H×M→ Λ thatrestricts to the pairing for cusp forms.

2.4.6. As we shall explain in a more canonical fashion in 2.7.6, there is an isomorphism Tquo ∼= S of h-modules given by Ohta’s Λ -adic Eichler-Shimura isomorphism [Oh1, Oh2]. Moreover, Ohta showed thatTsub ∼= h via a Λ -duality with Tquo.

2.5 Refining the method of Ribet and Mazur-Wiles: the map ϒ

We define the map ϒ of [Sh] and consider the relationship with the work of Mazur-Wiles [11]. Our descrip-tion is heavily influenced by the approaches of Wiles [26] and Ohta [Oh2].

We suppose that p≥ 5 and p -ϕ(N).4 We will work mostly in the θ -part (as in 2.2.4) for a fixed primitive,even character θ : (Z/N pZ)× → Qp

× such that the condition θω−1|(Z/pZ)× 6= 1 or θω−1|(Z/NZ)×(p) 6= 1of 2.2.4 holds. We also suppose that θ 6= ω2 in the case that N = 1.

2.5.1. We briefly outline the construction of ϒ : Yθ → Pθ that will appear in this section.We analyze the h[GQ]-action on Tθ/IθTθ , showing that it fits in an exact sequence

0→ Pθ → Tθ/IθTθ → Qθ → 0

of h[GQ]-modules. Any such exact sequence provides a cocycle GQ→Homh(Qθ ,Pθ ) that defines its exten-sion class in Galois cohomology. Our exact sequence has three key properties: the GL-action on Tθ/IθTθ

is unramified, the GL-actions on Pθ and Qθ are trivial, and the hθ/Iθ -module Qθ is free of rank 1 with acanonical generator. We may therefore modify our cocycle as follows. First, we compose it with evaluationat the generator of Qθ to obtain a map GQ→ Pθ . Since GL acts trivially on Pθ and Qθ , this map restricts toa homomorphism GL→ Pθ . Since the GL-action on Tθ/IθTθ is unramified, this homomorphism in turn fac-tors through a homomorphism X → Pθ . After a twist, it further factors through Yθ and provides the desiredmap ϒ : Yθ → Pθ , which we can show to be of Λ -modules.

We first explain that hθ/Iθ is the quotient of Λθ by a p-adic L-function ξθ . This will provide the connec-tion between the map ϒ and the Iwasawa main conjecture.

3 There is one potentially confusing aspect: the action of Λ → h on S⊂ΛJqK is not given by multiplication of the coefficientsof q-expansions by the element of Λ . It is instead this multiplication after first applying the inversion map λ 7→ λ ∗ on Λ thattakes group elements to their inverses.4 It should actually be possible to allow either or both of p = 3 and p | ϕ(N) in what follows.


2.5.2. We define the p-adic L-function ξθ .Note that any homomorphism G→Q× factors through some Gr and so induces an even Dirichlet char-

acter. Note also that G1 = (Z/N pZ)×/〈−1〉 and G∼= G1× (1+ pZp).The p-adic L-function ξθ is the unique element of Λθ that interpolates Dirichlet L-values at −1 in the

sense that for each character ε : G→Q× such that ε|G1 = θ , the ring homomorphism Λθ →Qp induced byε sends ξθ to the value L(ε−1,−1) ∈Q of the Dirichlet L-function.

We can also describe ξθ in terms of Kubota-Leopoldt p-adic L-functions. We make the identificationΛθ = Zp[θ ]JT K with T = [1+ p]−1, where [1+ p] is the group element of 1+ p ∈ 1+ pZp. We then havethe following equality of functions of s ∈ Zp:

ξθ ((1+ p)s−1) = Lp(ω2θ−1,s−1).

2.5.3. We construct a canonical isomorphism hθ/Iθ

∼−→Λθ/(ξθ ).Consider the ordinary Λ -adic Eisenstein series

Eθ = 12 (ξθ )

∗+∞

∑n=1

(∑d|n

(d,N p)=1

d[d]

)qn ∈Mθ ,

where [d] is the image in Λθ−1 of the group element in G for d, and λ 7→ λ ∗ is the involution defined in thefootnote of 2.4.5.5 By duality with the Hecke algebra, it provides a surjective homomorphism Hθ → Λθ ,the kernel of which is Iθ by definition.

Let MEis denote the component of an H-module M for the unique maximal ideal m containing the Eisen-stein ideal Iθ . By our choice of θ , the Eisenstein series Eθ is not congruent modulo m to any other Eisensteinseries [Oh3, Lemma 1.4.9]. It follows from this that the injection of S in M induces an exact sequence

0→SEis→MEis→Λθ → 0,

where the latter map takes a modular form to the (involution of the) constant term in its q-expansion. Ourmap πθ : hθ/Iθ → Λθ/(ξθ ) may then be constructed from the reduction of Eθ modulo ξθ . That is, Eθ is acusp form modulo (ξθ )⊆Λθ by the exact sequence, and this cusp form provides the surjective map πθ byduality with the Hecke algebra h. Once we know that πθ is an isomorphism, it is inverse to the map inducedby ιθ by definition.

We explain the idea behind the injectivity of πθ . We have an evident surjection Λθ → hθ/Iθ given bythe fact that every Hecke operator T (n) is identified modulo Iθ with an element of Λθ . So, hθ/Iθ is somequotient of Λθ . The Λ -adic forms in Mθ have integral constant coefficients, which can be seen by themethod of [Em, Proposition 1]. Given this, the existence of πθ is equivalent to the fact Eθ modulo (ξθ ) isa Λ -adic cusp form. As the constant coefficient of Eθ equals ξ ∗

θtimes a unit, no surjective homomorphism

to a larger quotient of Λθ can exist.6 Thus, πθ is an isomorphism, and it is inverse to the map induced by ιθ

by definition.

2.5.4. We define Qθ and construct a canonical surjection Tθ/IθTθ → Qθ of h[GQ]-modules.For a module M over a Λ -algebra h, let M ] denote the h[GQ]-module that is M as an h-module and

on which σ ∈ GQ acts through multiplication by the inverse of the image of σ in G. We then defineQθ = (hθ/Iθ )

](1). Consider the Jacobian variety Jr of the curve X1(N pr). Let Jr,tor ⊂ Jr(Q) be its tor-sion subgroup, and take the contravariant (i.e., dual) action of Tr on Jr,tor. Consider the class αr ∈ Jr(C) ofthe divisor (0)− (∞), where 0 and ∞ are viewed as cusps on X1(N pr)(C). It is torsion by the theorem ofDrinfeld and Manin [Dr1, Ma]. Moreover, αr is easily seen to be annihilated by Ir.

Let βr,θ be the image of αr in the θ -part of Jr[p∞] = Jr,tor⊗Zp. The Tr,θ -span Br,θ of βr,θ is a quotientof hr,θ/Ir,θ by definition. Moreover, Br,θ is isomorphic to Λr,θ/(ξr,θ ) by a computation of divisors of Siegelunits that says in particular that the θ -part of the divisor of g0, 1

N pris ξr,θ times (0)− (∞), up to a unit

5 The reader may wish to ignore the involutions in order to focus on the idea of the argument.6 Another, more usual, way to approach injectivity is to use Iθ + ξθhθ in place of Iθ until one recovers the equality of theseideals through a proof of the main conjecture, as in 2.5.7 below.


(see [11, Section 4.2]).7 Here, ξr,θ denotes the image of ξθ in Λr,θ . The GQ-action on Br,θ factors throughGal(Fr/Q), and we have σaβr,θ = 〈a〉−1βr,θ for any a ∈ (Z/N prZ)×.

Poincaré duality allows us to identify the first étale homology group of X1(N pr)/Q with the Tate twistof the first étale cohomology group. Taking this together with the canonical pairing of cohomology and thetorsion in Jr, we obtain a Galois-equivariant, perfect pairing

( , ) : Hét1 (X1(N pr)/Q,Zp)× Jr[p∞]→Qp/Zp(1)

with respect to which the Hecke operators are self-adjoint. Let ( , )θ denote the induced pairing on θ -parts.Define a map φ by

φ : Hét1 (X1(N pr)/Q,Zp)θ →Λr,θ ⊗Qp/Zp(1), x 7→ ∑

a∈Gr

[a]r⊗ (x,〈a〉βr,θ )θ ,

where [a]r ∈Λr,θ denotes the group element for a.8 Let ξr,θ be the image of ξθ in Λr,θ . As ξr,θ βr,θ = 0, theimage of the map φ is contained in the group (Λr,θ ⊗Qp/Zp(1))[ξr,θ ] of ξr,θ -torsion, and φ factors throughthe quotient Tr,θ/Ir,θTr,θ .

Consider the composition

Tr,θ/Ir,θTr,θφ−→ (Λr,θ ⊗Qp/Zp(1))[ξr,θ ]→ (Λr,θ/ξr,θ )(1)

ιr,θ−−→ (hr,θ/Ir,θ )(1),

where the second map is given by x 7→ ξr,θ x for any lifting x of x to Qp[Gr]θ (1). It is surjective by ourdescription of Br,θ and the perfectness of ( , )θ . As seen from the Galois action on βr,θ , it is moreover anhr[GQ]-module homomorphism Tr,θ/Ir,θTr,θ → (hr,θ/Ir,θ )

](1). The maps are compatible with r, and theirprojective limit is the desired surjective h[GQ]-module homomorphism Tθ/IθTθ → Qθ .

2.5.5. We explain how the surjection of 2.5.4 fits in an exact sequence


of h[GQ]-modules that is canonically locally split over GQp .We use the fact that the Eisenstein part T+

Eis→ Tquo,Eis of the canonical map of 2.4.4 is an isomorphism,or equivalently, that Tsub,Eis→ TEis/T

+Eis is an isomorphism. To see this, one uses an h-module splitting of

the local exact sequence for Tθ (see [Oh2]) and the method of Kurihara and Harder-Pink [Ku, HP]. We referthe reader to [FK, Section 6.3] for the argument.

Let us explain the use of this fact: by definition, complex conjugation acts on Qθ by multiplication by−1. Thus, Qθ is a quotient of Tθ/T

+θ

. By our isomorphism on Eisenstein components, it is a quotient ofTsub,θ/IθTsub,θ , which by 2.4.6 is isomorphic to hθ/Iθ as an h-module. This forces the quotient map to bean injection, so we have Qθ

∼= Tsub,θ/IθTsub,θ . But now, this tells us that Qθ is an h[GQp ]-submodule ofTθ/IθTθ . In other words, the surjection Tθ/IθTθ → Qθ is canonically locally split on GQp . We then havenecessarily that the kernel of the latter surjection is Tquo,θ/IθTquo,θ ∼= Pθ . This yields the exact sequence.

It is perhaps worth observing that this sequence is also identified with the reduction modulo Iθ of theexact sequence of h[GR]-modules 0→ T+

θ→ Tθ → Tθ/T

+θ→ 0. Finally, the determinant of the GQ-action

on Tθ is known (e.g., from the determinants of modular Galois representations) and agrees with the GQ-action on Qθ , so the GQ-action on Pθ is trivial.

2.5.6. We have that Pθ and Qθ have trivial actions of GL. Hence, we have a homomorphism

GL→ Homh(Qθ ,Pθ ), σ 7→ (x 7→ σ x− x),

where x is a lifting of x to Tθ/IθTθ . By 2.5.5, this homomorphism factors through the unramified quotient Xof GL. Thus, we have a homomorphism X→Homh(Qθ ,Pθ ) that is compatible with the action of Gal(L/Q).This gives a homomorphism of Gal(K/Q)-modules

X−(1)→ Homh(Qθ (−1),Pθ )∼= Homh((hθ/Iθ )],Pθ )∼= P [

θ ,

7 In the projective limit, this gives another way of defining the isomorphism hθ/Iθ

∼−→Λθ/(ξθ ).8 To make sense of this, note that the tensor product in the sum is taken over Zp[θ ].


where P [θ

is Pθ on which σa ∈ Gal(L/Q) acts as multiplication by 〈a〉−1. In other words, we have a Λθ -module homomorphism

ϒ : Yθ = X−(1)θ → Pθ ,

with the Galois action of G on the left and inverse diamond action of G on the right.

2.5.7. We describe the heart of the Mazur-Wiles proof of the Iwasawa main conjecture.The Iwasawa main conjecture is the equality of ideals

charΛθ(Yθ ) = (ξθ ).

By the analytic class number formula, this conjecture is reduced to charΛθ(Yθ )⊆ (ξθ ).

Let L be the h[GQ]-submodule of Tθ generated by Tsub,θ . It follows as in 2.5.5 that we have an equalityLm = Tsub,m⊕L+

m of Eisenstein components. Moreover, P′θ= L+/IθL

+ is GQ-stable in L/IθL. In otherwords, we have an exact sequence of h[GQ]-modules

0→ P′θ → L/IθL→ Qθ → 0.

In the same way as ϒ , we may define ϒ ′ : Yθ → P′θ

, which is now surjective by construction.The Iwasawa main conjecture can be deduced from this surjectivity of ϒ ′. More precisely, we use the

following facts:(3)

(1) The map ϒ ′ : Yθ → P′θ

is surjective.(2) We have that P′

θ= L+/IθL

+ with L+ a finitely generated, faithful hθ -module.(3) The kernel of the canonical surjection Λθ → hθ/Iθ is contained in (ξθ ).9

From (1), we obtaincharΛθ

(Yθ )⊆ charΛθ(P′θ ).

From (2) and (3), we can deduce that

charΛθ(P′θ )⊆ charΛθ

(Λθ/(ξθ )) = (ξθ ).

Hence charΛθ(Yθ )⊆ (ξθ ).

2.5.8. The conjecture stated in the following section implies that ϒ is surjective. This tells us that theinexplicit lattice L required for the Mazur-Wiles proof in 2.5.7 is precisely the canonical lattice Tθ . In thissense, it suggests a refinement of the method of Ribet and Mazur-Wiles.

2.6 The conjecture: ϖ and ϒ are inverse maps

We state the conjecture of the third author [Sh] and the result of the first two authors [FK].

2.6.1. In 2.2.4 and 2.5.6, we defined Λ -module homomorphisms

ϖ : Pθ → X−(1)θ and ϒ : X−(1)θ → Pθ .

We have the conjecture of the third author. See [Sh, Conjecture 4.12], where the conjecture is given up to acanonical unit; this stronger version was a stated hope of the third author.

Conjecture 1. The maps ϖ and ϒ are inverse to each other.

This conjecture provides an explicit description of X−(1)θ in terms of modular symbols. In this sense, itmay be viewed as a refinement of the main conjecture.

2.6.2. We state the result [FK, Theorem 7.2.3(1)] of the first two authors. Let ξ ′θ∈Λθ denote the derivative

of the p-adic L-function ξθ in the s-variable (see 2.5.2).

9 Actually, we know that the kernel coincides with (ξθ ), but this weaker statement is enough.


Theorem 1. We have ξ ′θϒ ϖ = ξ ′

θmodulo p-torsion in Pθ .

If ξθ has no multiple roots, the theorem implies the conjecture up to p-torsion in Pθ . In fact, it leads toproofs of the conjecture under various hypotheses: see [FK, Section 7.2].

2.6.3. McCallum and the third author conjectured that the image of the cup product

H1(Z[ζN pr , 1N p ],Zp(1))⊗Zp H1(Z[ζN pr , 1

N p ],Zp(1))∪−→ H2

ét(Z[ζN pr , 1N p ],Zp(2))

projects onto H2ét(Z[ζN pr , 1

N p ],Zp(2))+θ ([McS] for N = 1), which implies that ϖ is surjective. This genera-tion conjecture follows if we know that ϖ ϒ = 1. In particular, it holds if ξθ has no multiple roots, and italso holds if Pθ ⊗Zp Qp is generated by one element over Λθ ⊗Zp Qp [FK, Theorem 7.2.8].

2.7 The proof that ξ ′ϒ ϖ = ξ ′

We explain some of the important aspects of the proof of the main theorem, referring to the relevant sectionsof [FK] for details.

2.7.1. We consider a refinement of the diagram in 1.0.10 in which we divide the right-hand square of thatdiagram into two squares:

Sθ

z//

mod I

lim←−rH2

ét(Y1(N pr),Zp(2))θ

HS //

∞

H1ét(Z[

1N p ],Tθ (1))

reg//

Sθ

mod I

Pθ

ϖ // Yθ

ξ ′θ // Yθ

ϒ // Pθ .

Here, the maps z and ∞ are the projective limits of the θ -components of the maps zr and ∞r. The commu-tativity of the left square of the diagram in 2.7.1 is seen in Section 2.3. The discussion of the rest of thisdiagram, and the fact that the bottom row is also multiplication by ξ ′

θ, compose the rest of this subsection.

It is remarkable that ξ ′θ

appears here in two very different contexts. The ξ ′θ

that appears in the diagramand contributes to ξ ′

θϒ ϖ is related to cup product with the logarithm of the cyclotomic character. The

other ξ ′θ

is the constant term modulo ξθ of a Λ -adic modular form that appears in a computation of theregulators of zeta elements.

2.7.2. The map HS arises from the Hochschild-Serre spectral sequences

E i, j2 = H i(Z[ 1

N p ],Hj(Y1(N pr)/Q,Zp(2)))⇒ E i+ j = H i+ j(Y1(N pr),Zp(2)).

as the projective limit over r of maps E2→ E1,12 , followed by projection to the ordinary θ -part. We remark

thatH1

ét(Z[ 1N p ],Tθ (1))⊂ H1

ét(Z[ 1N p ], Tθ (1)),

and the image of HS is actually contained in the larger group, hence the dotted arrow. However, elementsof Sθ are carried to the smaller group under HSz [FK, Proposition 3.3.14], so we can still make sense ofthe diagram.

2.7.3. The third vertical arrow in the diagram of 2.7.1 is a composition of maps as follows:

H1ét(Z[ 1

N p ],Tθ (1))→ H1ét(Z[ 1

N p ],Qθ (1))∪(1−p−1) log(κ)−−−−−−−−−→ H2

ét(Z[ 1N p ],Qθ (1))

∼−→ Yθ .

The first map is induced by the surjection Tθ → Qθ . The map κ : GQ → Z×p is the p-adic cyclotomiccharacter, and log is the p-adic logarithm. The second map is the cup product, where we regard log(κ) =logκ as an element of H1

ét(Z[1p ],Zp).

For the third map, note that Qθ∼= (Λθ/ξθ )

](1) by 2.5.3 and 2.5.4. As the p-cohomological dimensionof Z[ 1

N p ] is 2, the group H2ét(Z[

1N p ],(Λθ/ξθ )

](2)) is isomorphic to the quotient of


H2ét(Z[ 1

N p ],Λ]

θ(2))θ

∼−→ lim←−r

H2ét(Or[

1N p ],Zp(2))θ

∼−→ Yθ

by the Gal(K/Q)-action of ξθ . Here, the first isomorphism is by Shapiro’s lemma, and the second is from2.2.4. By the main conjecture and the fact that Yθ has no finite Λ -submodules, Yθ is ξθ -torsion, so Yθ/ξθYθ =Yθ . Putting this all together, we have the map.

2.7.4. We define a functor D on pro-p GQp -modules.Let T = lim←−λ

Tλ for a projective system of finite abelian p-groups Tλ . For any abelian group M, setT ⊗M = lim←−λ

(Tλ ⊗M). Let W denote the Witt vectors of Fp, and suppose that the Tλ are endowed withcompatible actions of h[GQp ] for a pro-p ring h. We may then consider the h-module D(T ) that is the fixedpart

D(T ) = (T ⊗W )GQp

for the diagonal action of GQp on T ⊗W . If the GQp -actions on the Tλ are unamified, then D(T ) and Tare isomorphic h-modules. If T has trivial GQp -action, then D(T ) ∼= T ⊗W GQp = T ⊗Zp ∼= T , and thisisomorphism is canonical. See [FK, Section 1.7].

2.7.5. We define p-adic regulator maps for unramified, pro-p GQp -modules.Let T be as in 2.7.4, and suppose that the action of GQp on T is unramified. Let E = Q(W ) be the

maximal unramified extension of Qp. The p-adic regulator map [FK, Section 4.2]

regT : H1ét(Qp,T (1))→ D(T )

for T is the h-module homomorphism defined as the composition

H1ét(Qp,T (1))

inf−→ H1ét(E,T (1))

Frp=1 ∼−→ (T ⊗E×)Frp=1→ D(T ).

Here, the first map is inflation, the second is Kummer theory, and the final map is induced by

E×→W (Fp), x 7→ p−1 log( xp

Frp(x)

),

where the p-adic logarithm log is defined to take p to 0.Note that if GQp acts trivially on T , then regT is induced by the map (1− p−1) log : Q×p →Zp in a similar

fashion.

2.7.6. We define the p-adic regulator map reg in the diagram of 2.7.1.Note that Tquo has by definition an unramified GQp -action. We have a refinement [FK, Section 1.7] of

Ohta’s Λ -adic Eichler-Shimura isomorphism [Oh1]. That is, there is a canonical isomorphism of h-modulesD(Tquo)

∼−→S, and in particular Tquo and S are noncanonically isomorphic. The map reg is then defined asthe composition

reg : H1ét(Z[ 1

N p ],Tθ (1))→ H1ét(Qp,Tquo,θ (1))

regTquo−−−−→ D(Tquo)∼−→Sθ .

2.7.7. We explain the right-hand vertical map “mod I” in the diagram of 2.7.1.The GQp -action on Tquo/ITquo is trivial, so the canonical isomorphism D(Tquo)

∼−→ S provides an iso-morphism Tquo/ITquo

∼−→S/IS. In particular, we obtain “mod I” as the composition of projection followedby a string of canonical isomorphisms:

Sθ Sθ/IθSθ

∼−→ Tquo,θ/IθTquo,θ∼−→ T+

θ/IθT

+θ

∼−→ Sθ/IθSθ = Pθ .

2.7.8. The commutativity of the two right-hand squares in the diagram of 2.7.1 are nontrivial cohomologicalexercises. We mention only which calculations must in the end be performed.

(2)

(1) The commutativity of the middle square is reduced to that (see [FK, Section 9.4]) of


H1ét(Z[

1N p ],(Λθ/ξθ )

](2))∪(1−p−1) log(κ)

//

o

H2ét(Z[

1N p ],(Λθ/ξθ )

](2))

H2ét(Z[

1N p ],Λ

]θ(2))

ξ ′θ // H2

ét(Z[1

N p ],Λ]

θ(2)),

o

OO

the vertical arrows occurring in the long exact sequence in the Z[ 1N p ]-cohomology of

0→Λ]

θ(2)→Λ

]θ(2)→ (Λθ/ξθ )

](2)→ 0.

Thus, the ξ ′θ

that appears in the diagram is found in Galois cohomology.(2) The commutativity of the right-hand square is reduced to verifying that the map

Yθ∼= H2

ét(Z[ 1N p ],Qθ (1))← H2

ét(Z[ 1N p ],Tθ/IθTθ (1))→ H2

ét(Qp,Pθ (1))∼= Pθ

given by lifting and then projecting is well-defined and agrees with ϒ [FK, Section 9.5]. Here, the firstisomorphism was discussed in 2.7.3 and the last is the invariant map of local class field theory, recallingfrom 2.5.5 that Pθ has trivial Galois action. This description is closer to the construction of ϒ that willappear for Fq(t) in Section 3.

2.7.9. It remains to prove that the composition Sθ/IθSθ → Sθ/IθSθ → Pθ , where the first arrow is thecomposition of the upper horizontal arrows modulo Iθ in the diagram of 2.7.1, coincides with multiplicationby ξ ′

θon Pθ . This is deduced in [FK, Sections 4.3 and 8.1] from the computation of the p-adic regulators of

zeta elements given in [Oc, Fu]. This is a very delicate analysis: we explain only the rough idea of how ξ ′θ

appears at its end.The map Pθ → Pθ is shown to be given (modulo ξθ ) by multiplication by the constant term at t = 1 of a

p-adic L-function in a variable t that takes values in Mθ . This p-adic L-function is a product of two Λ -adicEisenstein series which vary with t. The constant term in the q-expansion of this product is itself a product oftwo zeta functions ζp(t)ξθ (s+ t−1), where ζp(t) is the p-adic Riemann zeta function and s is the variablefor Λθ ⊂ hθ . Note that ζp(t) has a simple pole at t = 1 with residue 1. To evaluate ζp(t)ξθ (s+ t−1) moduloξθ (s) at t = 1, we can first subtract ζp(t)ξθ (s) from the product and then take the resulting limit

limt→1

ξθ (s+ t−1)−ξθ (s)t−1

= ξ′θ (s).

In this manner, the map is shown to be multiplication by ξ ′θ

.

3 The case of Gl2 over Fq(t)

We now consider the field F = Fq(t) for some prime power q. In this section, we provide F-analogues ofthe constructions, conjecture, and theorem of Section 2. We require the following objects:

• the ring O= Fq[t],• the completion F∞ = F((t−1)) of F at the place ∞,• the valuation ring O∞ = FqJt−1K of F∞, which does not contain O,• a prime number p different from the characteristic of Fq,• a non-constant polynomial N ∈ O.

Let us also fix an embedding F → F∞ of separable closures. To avoid technical complications, we assumein this section that p does not divide (q+1)|(O/NO)×|.

The organization of this section follows closely that of Section 2. We hope to make clear that most con-structions are remarkably similar to the case of Q, though we also highlight differences. We work withcongruence subgroups of Gl2(O), rather than of SL2(Z). Modular symbols, used to construct ϖ , are nowfound in the homology S of the compactification of the quotient of the Bruhat-Tits tree by a congruence


subgroup. This S is a quotient of an étale homology group T of the Drinfeld modular curve, used in con-structing ϒ . As most constructions are so similar, we provide less detail than in Section 2. We intend for fulldetails to appear in a forthcoming paper.

3.1 From modular symbols to cup products: the map ϖ

3.1.1. We introduce homology groups S and M of the Bruhat-Tits tree.Consider the Bruhat-Tits tree B for PGL2(F∞). Its vertices are homothety classes of O∞-lattices L of rank

2 in F2∞ , or equivalently, elements of PGL2(F∞)/PGL2(O∞). This tree is (q+1)-valent, and two lattices L⊂

L′ connected by an edge if [L′ : L] = q.10 The oriented edges then correspond to elements of PGL2(F∞)/I∞,where I∞ is the Iwahori subgroup of matrices in PGL2(O∞) that are upper-triangular modulo the maximalideal of O∞. The group PGL2(F) acts on the left on B in the evident manner.

Let Γ1(N) be the congurence subgroup of Gl2(O) given by

Γ1(N) =

(a bc d

)∈ Gl2(O)

∣∣∣ (c,d)≡ (0,1) mod N.

We may complete the Bruhat-Tits tree to a space B∗ by adding in the (rational) ends, which correspond toelements of P1(F). We define

U(N) = Γ1(N)\B and U(N) = Γ1(N)\B∗.

The elements of Γ1(N)\P1(F) are the ends of U(N). Our homology groups, or spaces of modular symbols,are then

S= H1(U(N),Zp)⊂M= H1(U(N),ends,Zp).

3.1.2. We introduce Manin-Teitelbaum symbols [u : v] ∈M.Modular symbols in M were defined by Teitelbaum [Te] analogously to the case of Q. In particular,

given α,β ∈ P1(F), we have a modular symbol that is the class α → β of any non-backtracking path inthe Bruhat-Tits tree that connects the two corresponding ends of B.

Analogues of Manin symbols are defined as before. That is, for u,v∈O/NO with (u,v) = (1), we chooseγ =

(a bc d

)∈ Gl2(O) with u = c mod N and v = d mod N, and then

[u : v] =

dbN→ c

aN

.

These symbols generate H and yield a presentation with identical relations to those of 2.1.2.

3.1.3. We introduce the intermediate space M0 on which we define ϖ .Let M0 denote the Zp-submodule of M generated by the Manin symbols [u : v] with u,v 6= 0. As in the

case of Gl2 over Q, we have S⊂M0 ⊂M.

3.1.4. We introduce cyclotomic N-units λ uN

in abelian extensions FN ⊂ EN of Fq(t). The reader may find apowerful analogy with objects in the theory of cyclotomic fields over Q.

We consider the cyclotomic N-units λ uN

for u ∈ O− (N). These are the roots of the Carlitz polynomials[Ca] for divisors of N, or are equivalently the N-torsion points of the Carlitz module. As λ u

Ndepends only

on u modulo N, we abuse notation and consider it for nonzero u ∈ O/NO. We can visualize λ uN

in thecompletion C∞ of F∞ by

λ uN= exp

(uπ

N

)=

uN ∏

a∈O−0

(1− u

Na

),

where exp is the Carlitz exponential and π ∈ C∞ is transcendental over F .Let EN = F(λ 1

N), which is an abelian extension of F of conductor N∞ containing no constant field

extension of F . There is an isomorphism Gal(EN/F)∼−→ (O/NO)× such that a∈ (O/NO)× is the image of an

10 Note that q appears in this sentence as the order of the residue field of O∞.


element σa ∈Gal(EN/F) that satisfies σa(λ 1N) = λ a

N. Let FN be the largest subfield of EN in which ∞ splits

completely over F , which we might call the ray class field of modulus N. Under the above isomorphism,Gal(EN/FN) is identified with F×q . In fact, we have σc(λ u

N) = cλ u

Nfor c ∈ F×q . These facts are found in the

work of Hayes [Ha].Let ON denote the integral closure of O in FN . Since p - (q− 1) by assumption, the image of λ u

Nin the

p-completion of the N-units of EN is fixed by the action of F×q . This allows us to view λ uN

as an element ofH1

ét(ON [1N ],Zp(1)). For nonzero u,v ∈ O/NO, we may consider the cup product

λ uN∪λ v

N∈ H2

ét(ON [1N ],Zp(2)).

3.1.5. We define the map ϖ . Here, we work directly with étale cohomology, rather than K2.There is a homomorphism

ϖ : M0→ H2ét(ON [

1N ],Zp(2)), [u : v] 7→ λ u

N∪λ v

N.

In the current setting, we can no longer quickly verify from the presentation of M0 that ϖ is well-defined.Rather, we see this as a consequence of the argument that ϖ is “Eisenstein” in Section 3.3.

3.1.6. We introduce the cuspidal Hecke algebra h and its Eisenstein ideal I.Let n denote a nonzero ideal of O. Through the action of Gl2(F) on B, we have a Hecke operator T (n)

acting on S as the correspondence associated to Γ1(N)(

1 00 n

)Γ1(N), where n = (n). Let h be the subring of

EndZp(S) generated over Zp by the Hecke operators T (n).We also have diamond operators 〈a〉 in h for nonzero ideals a of O prime to (N). This 〈a〉 depends only

on the reduction modulo N of the monic generator of a.The Eisenstein ideal I is the ideal of h generated by T (n)−∑d|nN(d)〈d〉 for all nonzero ideals n of O,

taking 〈d〉= 0 if d+(N) 6= (1). Here, N(n) = [O : n] is the absolute norm of n.Similarly, we have the Eisenstein ideal I of the Hecke algebra H⊂ EndZp(M).

3.1.7. To say that ϖ is “Eisenstein” is to say that ϖ factors through a map

ϖ : M0/IM0→ H2ét(ON [

1N ],Zp(2)).

We explain this result in Section 3.3.

3.1.8. Let G = (O/NO)×/F×q , and set Λ = Zp[G].(2)

(1) We have a ring homomorphism ι : Λ → h which sends the group element [a] in Zp[G] for a ∈ G to theinverse 〈a〉−1 of the diamond operator corresponding to a.

(2) We have the isomorphism Gal(FN/F)∼−→ G of class field theory (see 3.1.4).

Modules over h and Zp[Gal(FN/F)] become Λ -modules through these identifications.

3.2 Working with fixed level

We explain why we work with fixed level in Section 3, and we define our two objects of study.

3.2.1. We do not pass up a tower for the following reason on the Gl1-side. By assumption on p, the field Fqhas no nontrivial pth roots of unity. Since FN/F contains no constant field extension, FN also contains nonontrivial pth roots of unity. So, even if we “increase” N, we are unable to employ the Iwasawa-theoretictrick of passing Tate twists through projective limits of Galois cohomology groups. In particular, since wedeal with cohomology with Zp(2)-coefficients, we do not work with class groups.

3.2.2. We again have two objects of study:

• the geometric object P = S/IS for Gl2,• the arithmetic object Y = H2

ét(ON ,Zp(2)) for Gl1.


Given a character θ : G→ Qp×, we set Λθ = Zp[θ ] and view it as a quotient of Λ through θ . For a

Λ -module M, we let Mθ = M⊗Λ Λθ denote the θ -part of M. If θ is primitive, then our assumption that pdoes not divide |G| implies that the canonical maps

Pθ →M0θ/IθM

0θ and Yθ → H2

ét(ON [1N ],Zp(2))θ

are isomorphisms.

3.3 Zeta elements: ϖ is “Eisenstein”

We explain that ϖ factors through the quotient of M0 by the Eisenstein ideal I.

3.3.1. We define Siegel units on Drinfeld modular curves.Let Y (N) denote the Drinfeld modular curve that is the moduli scheme for pairs consisting of a rank 2

Drinfeld module over an O[ 1N ]-scheme and a full N-level structure (or, basis of the N-torsion) on it. Over

Y (N), we have a universal Drinfeld module, equipped with a full N-level structure, which locally lookslike (N−1O/O)2×Y (N). On the universal Drinfeld module is a certain theta function Θ . Given an elementof ( u1

N , u2N ) ∈ (N−1O/O)2, we may pull Θ back to a unit on the Drinfeld modular curve using the second

coordinate of the level structure. This unit gα,β ∈ O×Y (N)is the analogue of a Siegel unit.11

Let Y1(N) be the moduli scheme for pairs consisting of a rank 2 Drinfeld module over an O[ 1N ]-scheme

and a point of order N on it. If we take α = 0, then the Siegel unit g0,β may again be viewed as an elementof O×Y1(N)

.

3.3.2. If we take a K-theoretic product of two Siegel-type units, we obtain the Beilinson-type elementsconsidered by Kondo and Yasuda [KY]. See also the work of Kondo [Ko] and Pal [Pa]. Much as in the caseof Q, we have a map

z : M0→ H2ét(Y1(N),Zp(2)), [u : v] 7→ g0, u

N∪g0, v

N

of H-modules. We can specialize this at the cusp corresponding to ∞ ∈ P1(F) to obtain λ uN∪ λ v

N. This

specialization map ∞ is Eisenstein. Hence, we see that

ϖ = ∞ z : M0→ H1ét(ON [

1N ],Zp(2))

is well-defined and Eisenstein.

3.4 Homology of Drinfeld modular curves

In this subsection, we study the étale homology groups of Drinfeld modular curves. Unlike in Section2.4, we do not take ordinary parts. That is, the Galois representations found in the homology of Drinfeldmodular curves are already “special at ∞,” the required analogue of “ordinary at p.” Moreover, the resultingunramfied-at-∞ quotient may in the present setting be identified with the space S of cuspidal symbols, whichis the analogue of the plus quotient of homology of 2.2.2. In other words, the place ∞ of Fq(t) plays boththe roles that p and the real place do in the Gl2-setting over Q.

The statements in this subsection are consequences of the work of Drinfeld [Dr2].

3.4.1. We first introduce the étale homology group T.Over F , the Drinfeld modular curve Y1(N)/F has a smooth compactification X1(N)/F . Over C∞ (or F),

it is given by adding in the set of cusps Γ1(N)\P1(F) of the Drinfeld upper half-plane. We define our étalehomology group

T = Hét1 (X1(N)/F ,Zp)

11 Actually, gα,β as we have described it is not well-defined until we take its q2−1 power. The assumption that p - (q2−1) isused to avoid this issue when we work with étale cohomology.


as the Zp-dual of H1ét(X1(N)/F ,Zp).

The Hecke algebra generated by the T (n) in EndZp(T) is in fact equal to h. The module Qp⊗Zp T overthe total quotient ring Qp⊗Zp h of h is free of rank 2.

3.4.2. We study the action of GF∞on T.

We have an exact sequence0→ Tsub→ T→ Tquo→ 0

of h[GF∞]-modules, with Tsub and Tquo defined as follows. First, Tsub is the largest submodule of T such

that GF∞acts on Tsub(−1) trivially, and Tquo is the quotient. Then Tquo is equal to the maximal unramified,

h-torsion-free quotient of T. In this way, the place ∞ plays the role that the place at p does in 2.4.4. In fact,GF∞

acts trivially on Tquo, and both Qp⊗Zp Tsub and Qp⊗Zp Tquo are free of rank 1 over Qp⊗Zp h.The above short exact sequence is split as a sequence of h-modules: Tquo is the isomorphic image of the

h-submodule of T on which a choice of Frobenius element acts trivially. This will be used in constructingϒ below.

3.4.3. Since U(N) is essentially the graph of the special fiber of a model of X1(N) over O∞, we have asurjective homomorphism

T→ S= H1(U(N),Zp).

Via this map, S is identified with the quotient Tquo of T with trivial GF∞-action. In this way, Tquo is also

analogous to the plus quotient of homology in 2.2.2. That is, the place ∞ also plays the role that the realplace does over Q.

3.4.4. Let S be the space of those Zp-valued, special-at-∞ cuspidal automorphic forms

φ : PGL2(F)\PGL2(AF)/(K f × I∞)→ Zp,

where AF = A fF ×F∞ is the adele ring, K f is the closure of the image of Γ1(N) in PGL2(A f

F) (see 4.1.3),and I∞ is the Iwahori subgroup of PGL2(F∞). For φ to be special at ∞ means that its right Qp[Gl2(F∞)]-spanis a direct sum of copies of the “special representation.” (The latter is the quotient of the locally constantfunctions P1(F∞)→Qp by the constant functions.)

The property of being special at ∞ tells us the local behavior at the prime ∞ of the 2-dimensional Qp-Galois representation attached to the cusp form. This is a replacement for the condition of ordinarity at p: itis what tells us the GF∞

-action on T used in 3.4.2.

3.4.5. We explain how the groups S and S may be identified.The identification passes through the harmonic cocycles on U(N). These are the functions on the oriented

edges of U(N) that change sign if we switch its orientation of an edge and which sum to zero on the edgesleading into a vertex (i.e., are harmonic). The cuspidal harmonic cocycles are those supported on finitelymany edges. The space of Zp-valued cuspidal harmonic cocycles may be directly identified with S. It alsoprovides a combinatorial description of S. To see this, one starts with the observation that the double cosetspace on which forms in S are defined is none other than the set of oriented edges of U(N). The propertyof being special at ∞ gives the harmonic condition, and the two notions of cuspidality coincide. Thus, thespaces S and S that appear in the diagram of 1.0.10 are canonically identified in the case of Fq(t).

3.5 The map ϒ

We define the map ϒ : Yθ → Pθ on θ -parts for a fixed primitive character θ : G→Qp×.

3.5.1. We briefly outline the construction of ϒ : Yθ → Pθ .As in 2.5.1, we analyze the h[GF ]-action on Tθ/IθTθ , showing that it fits in an exact sequence


of h[GF ]-modules. Similarly to the setting of Gl2 over Q, the GF -actions on Pθ and Qθ are understood, andQθ is free of rank 1 over hθ/Iθ with a canonical generator. However, the domain of our map ϒ is not a


Galois group, so our approach to constructing ϒ is different. We employ compactly supported cohomology,which is dual to Galois cohomology by Poitou-Tate duality. Instead of directly using the cocycle attached tothe exact sequence, we construct ϒ in 3.5.7 from a connecting homomorphism ∂ on compactly supportedétale cohomology that appears as the second map in a composition of Λθ -module homomorphisms

ϒ : Yθ

∼−→ H2c,ét(O[

1N ],Qθ (1))

∂−→ H3c,ét(O[

1N ],Pθ (1))

∼−→ Pθ .

The isomorphisms are seen using the hθ [GF ]-module structure of Qθ and the triviality of the GF -action onPθ , respectively.

3.5.2. We define the L-function for θ by

L(θ ,s) = ∏p-N

(1−θ(Frp)−1N(p)−s)−1,

where the product is taken over the prime ideals p of O not dividing N, and Frp denotes an arithmeticFrobenius at p. We then take ξθ ∈Λθ to be the nonzero value L(θ ,−1).

3.5.3. We have an isomorphism hθ/Iθ

∼−→Λθ/(ξθ ). We indicate one construction of the map.Consider the Jacobian variety J of X1(N) and the class α ∈ J(C∞) of the divisor (0)− (∞), where 0 and

∞ are cusps on the Drinfeld modular curve. Gekeler showed that α has finite order [Ge], and it is annihilatedby I. The hθ -module generated by the θ -part of α is Λθ/(ξθ ) by a computation of the divisors of Siegelunits, providing the desired map.

3.5.4. We define Qθ = (hθ/Iθ )](1), where ( )] indicates a GF -action under which any element that maps

to a ∈ G acts by multiplication by θ−1(a). Much as in 2.5.4, pairing with the θ -part of α gives rise to acanonical surjection of hθ [GF ]-modules Tθ/ITθ → Qθ .

3.5.5. The exact sequence0→ Pθ → Tθ/IθTθ → Qθ → 0

of h[GF ]-modules is constructed as in 3.5.1. Here, we observe that Qθ has a nontrivial action of the Frobe-nius element chosen in 3.4.2, so Qθ is a quotient of Tsub. As before, Tsub is Zp-dual to Tquo and therebyisomorphic to h, so we have an isomorphism Tsub,θ/IθTsub,θ

∼−→ Qθ that provides a GF∞-splitting of the

exact sequence. The known GF -action on Qθ and the known determinant of the GF -action on Tθ tell us thatGF acts trivially on Pθ .

3.5.6. The analogue of the Iwasawa main conjecture over FN is the equality

|Yθ |= [Λθ : (ξθ )]

of orders. This equality is a consequence of Grothendieck trace formula, so we do not require the methodof Mazur-Wiles to prove it.

3.5.7. We define our map ϒ .Let H i

ét,c(O[1N ],M) denote the ith compactly supported étale cohomology group of a compact ZpJGFK-

module M that is unramified outside N∞. These groups fit in a long exact sequence

· · · → H iét,c(O[

1N ],M)→ H i

ét(O[1N ],M)→

⊕v|N∞

H iét(Fv,M)→ H i+1

ét,c (O[1N ],M)→ ··· .

The exact sequence in 3.5.1 yields a connecting homomorphism

H2ét,c(O[

1N ],Qθ (1))→ H3

ét,c(O[1N ],Pθ (1)) = Pθ ,

the latter identification as Pθ has trivial GF -action, and we can prove that the canonical map

H2ét,c(O[

1N ],Qθ (1))→ H2

ét(O[1N ],Qθ (1))

is an isomorphism. Hence, the above homomorphism is identified with H2ét(O[

1N ],Qθ (1))→ Pθ . Our ϒ is

defined as the following composition:


ϒ : Yθ

∼−−→ H2ét(ON [

1N ],Zp(2))θ

∼−−→ H2ét(O[

1N ],Λ

]θ(2))

∼−−→ H2ét(O[

1N ],(Λθ/ξθ )

](2))∼−−→ H2

ét(O[1N ],Qθ (1))−→ Pθ .

The isomorphism in the first line is by 3.2.2, the isomorphism in the second line is by Shapiro’s lemma, theisomorphism in the third line follows from the fact that ξθ kills Yθ by 3.5.6, and the isomorphism in thefourth line is by definition of Qθ in 3.5.4.

3.6 The conjecture: ϖ and ϒ are inverse maps

We state the conjecture and our main result in the case of Gl2 and Gl1 over Fq(t).

3.6.1. We state the conjecture.

Conjecture 2. The maps ϖ : Pθ → Yθ and ϒ : Yθ → Pθ are inverse to each other.

3.6.2. We state the theorem.

Theorem 2. We have that ξ ′θϒ ϖ = ξ ′

θ, where

ξ′θ =

ddq−s L(θ−1,s)|s=−1 ∈Λθ .

3.6.3. We can prove the order of Pθ is divisible by the order of Λθ/(ξθ ) and hence by the order of Yθ . Thus,in the case that ξ ′

θis a unit in Λθ , our conjecture is implied by the above theorem.

3.7 The proof that ξ ′ϒ ϖ = ξ ′

The method of the proof of our Theorem 2 is parallel to the proof in the Q-case. We give only its bareoutline.

3.7.1. As in 2.7.1, we consider a refinement of the diagram in 1.0.10 in which we divide the right-handsquare of that diagram into two squares:

Sθ

z//

mod I

H2ét(Y1(N),Zp(2))θ

HS //

∞

H1ét(ON [

1N ],Tθ (1))

reg//

Sθ

mod I

Pθ

ϖ // Yθ

ξ ′θ // Yθ

ϒ // Pθ .

The commutativity of the leftmost square of the diagram was discussed in Section 3.3. We discuss the mapsin the other two squares of the diagram below.

3.7.2. The map HS in the diagram arises in a Hochschild-Serre spectral sequence. An analogue of thediscussion of 2.7.2 applies.

3.7.3. Let κ be the canonical generator of H1ét(Fq,Zp). The third vertical arrow in the diagram is the com-

positionH1

ét(O[1N ],Tθ (1))→ H1

ét(O[1N ],Qθ (1))

∪κ−−→ H2ét(O[

1N ],Qθ (1))

∼−→ Yθ ,

where the last isomorphism is given in 3.5.7.


3.7.4. The map reg in the diagram is the θ -part of the p-adic regulator map

H1ét(F∞,S(1))

∪κ−−→ H2ét(F∞,S(1))

∼−→ S=S,

where the second map is the invariant map of local class field theory.

3.7.5. Since S and S are canonically identified, both “mod I” maps in the diagram are just reduction moduloIθ .

3.7.6. The proofs of the commutativity of the other two squares are once again nontrivial, though slightlydifferent, exercises in étale and Galois cohomology.

3.7.7. It remains to prove that the composition Sθ →Sθ → Pθ , where the first arrow is the composition ofthe upper horizontal rows, coincides with ξ ′

θtimes the reduction modulo Iθ map Sθ → Pθ . By the computa-

tion of Kondo-Yasuda [KY] of the values of a regulator map on the analogues of Beilinson elements, this isreduced to a comparison of their regulator map with the above p-adic regulator map.

4 What happens for Gld?

In this section, we discuss three settings for the study of generalizations of the conjectures in Sections 2 and3 for Gld over a field F , for a fixed integer d ≥ 1. The fields F and, thereby, the cases we consider here are:(iii)

(i) the rational numbers,(ii) an imaginary quadratic field,(iii) a function field in one variable over a finite field.

We have results only in the cases (i) and (iii) for d = 2 discussed above, but we wish to speculate andpose questions in a more general setting. Rather than formulating precise conjectures, we aim for the moremodest goals of pointing in their direction and inspiring the reader to investigate further.

4.1 The space of modular symbols

4.1.1. By an infinite place, we mean the unique archimedean place in cases (i) and (ii) and a fixed place ∞

in case (iii). The remaining places are called finite places. We have the following objects:

• the subring O of F of elements that are integral at all finite places,• the completion Fv of F at a place v,• the valuation ring Ov of Fv at a nonarchimedean place v,• the adele ring AF of F and the adele ring A f

F of finite places.• the subring O

fA = ∏v finiteOv of A f

F .

In the discussion below, we will use the notation ( )(d) when defining an object in the Gld-setting and thenomit the notation in many instances in which d is clear.

4.1.2. We define a topological space Dd by using the standard maximal compact subgroup of PGLd(F∞):in the respective cases, it is (iii)

(i) PGLd(R)/POd(R), so that SLd(R)/SOd(R)∼−→ Dd ,

(ii) PGLd(C)/PUd , so that SLd(C)/SUd∼−→ Dd ,

(iii) the Bruhat-Tits building associated to PGLd(F∞).

For example, in case (i) the space D2 is the complex upper half-plane H. In case (ii), the space D2 is thethree-dimensional hyperbolic upper-half space H3. Note that in case (iii), the Bruhat-Tits building has theset PGLd(F∞)/PGLd(O∞) of homothety classes of O∞-lattices of rank d in Fd

∞ as its 0-simplices.


4.1.3. Let N be a nonzero ideal of O. Let K(d)1 (N) be the open compact subgroup of Gld(O

fA) given by

K(d)1 (N) =

g ∈ Gld(O

fA) | (gd,1, . . . ,gd,d−1,gd,d)≡ (0, . . . ,0,1) mod N

.

LetU (d)(N) = Gld(F)\(Gld(A f

F)/K(d)1 (N)×Dd).

The space U (1)(N) is the relative Picard group Pic(O,N), viewed as a discrete space. For d ≥ 2, the spaceU (d)(N) is homeomorphic to the disjoint union of |Pic(O)| copies of Γ

(d)1 (N)\Dd , where

Γ(d)

1 (N) = Gld(O)∩K(d)1 (N).

4.1.4. Consider cases (i) and (ii). Let ε ∈ Gld(O) be a diagonal matrix with entries a1,a2, . . . ,ad such thatthe product a1a2 · · ·ad generates the roots of unity µF = O× in F . Let

Γ(d)

1 (N) = Γ(d)

1 (N)∩SLd(O).

Then Γ1(N)\Dd is identified with the quotient of Γ1(N)\Dd by the action of the operator

class(g) 7→ class(εgε−1)

for g ∈ SLd(R) in case (i) and for g ∈ SLd(C) in case (ii).

4.1.5. In case (i), the space U (2)(N) is identified with the quotient of Y1(N)(C) = Γ1(N)\H by the actionof complex conjugation on Y1(N)(C). In fact, the description in 1 shows that it arises from the quotient ofH by the action

H 3 x+ iy = class(√

y x0 1/

√y

)7→ class

((−1 00 1

)(√y x

0 1/√

y

)(−1 00 1

))= class

(√y −x

0 1/√

y

)=−x+ iy,

which coincides with the action of complex conjugation on Y1(N)(C).4.1.6. The space S(d)(N) of (cuspidal) modular symbols for Gld is defined as

S(d)(N) = image(Hd−1(U (d)(N),Z)→ HBMd−1(U

(d)(N),Z)),

where HBM∗ denotes Borel-Moore homology. Recall that if U(N) is a compactification of U(N), then

HBMi (U(N),Z) is canonically isomorphic to the relative homology group Hi(U(N),U(N) \U(N),Z). The

space S(N) may be the homology group Hd−1(U(N),Z) for some good choice of compactification.

4.1.7. In case (i), we have by 4.1.5 a canonical map

H1(X1(N)(C),Z)+→ S(2)(N) = H1(U(N),Z),

where U(N) is the quotient of X1(N) by the action of complex conjugation. This map is a surjection with2-torsion kernel.

4.1.8. For a nonzero ideal n of O, let T (n) denote the Hecke operator on S(d)(N) corresponding to the sumof K(d)

1 (N)-double cosets of elements of Md(OA)∩Gld(A fF) with determinant generating nOv in the vth

component. (For d = 1, we make the convention that T (n) = 0 if n and N are not coprime.) These operatorssatisfy T (ab) = T (a)T (b) for coprime a and b.

Let T(d)(N) denote the commutative subring of EndZ(S(d)(N)) generated by the T (n) with n a nonzeroideal of O.

4.1.9. For d = 1, the group S(N) of modular symbols is H0(U(N),Z) = Z[Pic(O,N)]. The Hecke algebraT(N) is the ring Z[Pic(O,N)], with T (n) for n coprime to N equal to the group element for n. Under theseidentifications, T(N) acts by left multiplication on S(N).


4.1.10. The modular symbol 0→∞ in Section 2.1 is generalized to the following element of HBMd−1(U(N),Z).

It is the class of the image in the identity component of U(N) of the following standard subset of Dd , witha suitable orientation: (i-ii)

(i-ii) the set of classes of diagonal matrices in Gld(F∞) with positive real entries,(iii) the union of all (d− 1)-simplices with 0-vertices in the set of classes in Dd of diagonal matrices in

Gld(F∞).

The modular symbols α → β for α,β ∈ P1(Q) are generalized to the classes in HBMd−1(U(N),Z) of the

images in U(N) of the translations by Gld(F) of the above standard subset of Dd .

4.2 Questions for the general case

We suspect that our results in Sections 2 and 3 are special cases of a relationship

Modular symbols for Gld modulo the Eisenstein ideal⇐⇒ Iwasawa theory for Gld−1

that holds for d ≥ 2. In this subsection, we describe what we expect to be true.

4.2.1. We lay out some basic objects, starting with:

• a prime number p 6= charF ,• a nonzero ideal N of O that is coprime to p,• a commutative pro-p ring R and its total ring of quotients Q(R),• a profinite R-module T with a continuous R-linear action of GF that is unramified at every finite place

not dividing N p,

Recalling from 4.1.8 the Hecke algebra T(d)(N) and modular symbols S(d)(N), we define

TR = lim←−r(R⊗T(d)(N pr)) and SR = lim←−

r(R⊗S(d)(N pr)).

We shall often use the fact that (p) = O in case (iii). For instance, in this case N pr = N, so we have quitesimply that TR = R⊗T(d)(N) and SR = R⊗S(d)(N).

We also let T(d)(N pr)′ be the subring of T(d)(N pr) generated by the T (n) with n coprime to (p). Notethat T(d)(N pr)′ = T(d)(N) in case (iii) and T(1)(N pr)′ = T(1)(N pr) in all cases.

4.2.2. We place some conditions on the pair (R,T ): (2)

(1) The Q(R)-module V = Q(R)⊗R T is free of rank d−1.(2) For every prime ideal p of O that does not divide N p, the characteristic polynomial Pp(u) = detQ(R)(1−

Fr−1p u |V ) of an arithmetic Frobenius Frp lies in R[u].

For a prime ideal p of O that does not divide N p, we define a(pn) for n≥ 0 by

Pp(u)−1 =∞

∑n=0

a(pn)un ∈ RJuK.

We then suppose: (3)

(3) There exists a ring homomorphism

φT : lim←−r(Zp⊗T(d−1)(N pr)′)→ R

that sends T (pk) to a(pk) for all prime ideals p of O not dividing N p and all k ≥ 1.

We extend a to a function on all nonzero ideals n of O by setting a(n) = φT (T (n)) if n is coprime to p anda(n) = 0 otherwise. In the case d = 2, our definition forces a(n) = 0 for any n not coprime to N p, while ingeneral, these values of a may not be uniquely determined by T , so φT should be considered as part of thedata.


4.2.3. We define the Eisenstein ideal IT of TR to be the ideal of the Gld-Hecke algebra TR generated by theelements

T (n)−∑d|n

a(d)N(d)

for the nonzero ideals n of O. Note that IT depends only on V and the choice of φT , rather than T itself. Incase (i), the ideal IT is generated by the coefficients of the formal expression

∞

∑n=1

T (n)n−s−ζ (s)∞

∑n=1

(n,p)=1

a(n)n−(s−1).

4.2.4. For any compact RJGFK-module M that is unramified outside of S ∪ ∞ for some finite set Sof finite places of F including those dividing p, we denote more simply by H2

ét(O[1p ],M) the R-module

H2ét(O[

1p ], j∗M), where j : Spec(O) \ S → Spec(O[ 1

p ]) is the inclusion morphism. It is independent of thechoice of S. We will also use a similar notation with O replaced by its integral closure in a finite extensionof F .

4.2.5. Our two objects of study are the R-modules:

• the geometric object P = SR/ITSR on the Gld-side,• the arithmetic object Y = H2

ét(O[1p ],T (d)) on the Gld−1-side.

We ask a vague question.

Question 1. Under what conditions does there exist a canonical isomorphism ϖ : P ∼−→ Y of R-modules?

We remark that there certainly must be some conditions, as different lattices T in V may have Y that arenonisomorphic. In what follows, we introduce three settings for further study.

4.2.6. We fix some notation for abelian extensions of F and their Galois groups.For r≥ 0, let Hr be the ray class field of F of modulus (pr), and let Or be the integral closure of O in Hr.

Let Γr = Pic(O,(pr)), which is canonically isomorphic to Gal(Hr/F) by class field theory. Let Γ = lim←−rΓr.

• In case (i), we have that Hr =Q(µpr)+, Or = Z[ζpr ]+, and Γ = Z×p /〈−1〉.• In case (ii), the field Hr is generated over F by the j-invariant j(E) and x-coordinates of the pr-torsion

points of an elliptic curve E over F( j(E)) with CM by O. There is an exact sequence

0→ (Zp⊗O)×/µF → Γ → Pic(O)→ 0.

Note that Γ /Γtor ∼= Z2p, where Γtor is the torsion subgroup of Γ .

• In case (iii), we have that Hr = H0, Or = O0, and Γ = Γr = Pic(O).

Let [a] ∈ ZpJΓ K be the group element corresponding to a ∈ Γ . We may also speak of [a] for a an idealof O coprime to p by taking the sequence of classes of a in the groups Γr. We use ( )] below to denote the(additional) GF -action on a module over a ZpJΓ K-algebra under which an element that restricts to a ∈ Γ

acts by multiplication by [a]−1.

4.2.7. We describe setting (Ad) for d ≥ 2.Let R0 be the valuation ring of a finite extension K0 of Qp. Let T0 be a free R0-module of rank d− 1

endowed with a continuous R-linear action of GF . We assume that the GF -action on T0 is unramified at allfinite places not dividing N p. We suppose that condition (3) of 4.2.2 is satisfied for (R0,T0), and we usea0(n) to denote a(n) of 4.2.2 for this pair.

Let R = R0JΓ K and T = R]⊗R0 T0. Then the pair (R,T ) satisfies conditions (1) and (2) of 4.2.2, and wesuppose that it satisfies (3). It follows directly that a(n) = [n]−1⊗a0(n) for any nonzero ideal n of O that iscoprime to N p. By definition of T , we also have an R-module isomorphism

Y = H2ét(O[

1p ],T (d))

∼= lim←−r

H2ét(Or[

1p ],T0(d)).

In that the GF -stable R0-lattice T0 has not been chosen with any special properties inside V0 = K0⊗R0 T0,we consider an additional condition.

(4)


(4) The GF -representation k0⊗R0 T0 is irreducible over the residue field k0 of K0.

It follows from (4) that the isomorphism class of T0 as an R0[GF ]-module depends only on the K0-representation V0 of GF . That is, all GF -stable R0-lattices in V0 have the same isomorphism class. Hence,the isomorphism class of the R-module Y depends only on V0.

Finally, to avoid known exceptions in case (i), we consider a primitivity condition.(5)

(5) The map φT0 does not factor through lim←−r(Zp⊗T(d−1)(Mpr)′) for any ideal M of O properly containing

N.

4.2.8. We may now ask our question for setting (Ad) under conditions (1)–(5).

Question 2. Does there exist a canonical isomorphism ϖ : P ∼−→ Y of R-modules?

We are also interested in what happens if conditions (4) and (5) are removed. For instance, we wonder if (5)might be removed for good choices of N, p, and d, or if (4) might be removed in the presence of a good,canonical lattice T0. In any case, we can ask the following question.

Question 3. If we do not suppose conditions (4) and (5), does there still exist a canonical isomorphismϖQp : Qp⊗Zp P ∼−→Qp⊗Zp Y ?

4.2.9. The p-adic Galois representations V0 attached to the following objects of conductor N pr for somer ≥ 0 all have (R0,T0) and (R,T ) satisfying (1)–(3):

• in case (i) for d = 2, an even Dirichlet character,• in case (i) for d = 3, a holomorphic cupsidal eigenform,• in case (ii) for d = 2, an algebraic Hecke character on A×F ,• in case (iii) for d ≥ 2, a cuspidal eigenform of Gld−1 that is special at ∞.

The examples for d = 2 obviously satisfy (4), and in the remaining cases, (4) may be assumed. By takingeach of the objects to be primitive, we may assume (5).

4.2.10. We explain how the setting (A2) for F =Q and Fq(t) was studied in Sections 2 and 3.Let θ be a primitive character of Pic(O,N p), and impose all the assumptions on p, N, and θ of Sections

2 and 3. Take R0 = Zp[θ ], and let T0 = Zp[θ ] with GF acting through θ−1. Let ∆ = Pic(O,(p)), which wemay view as a subgroup of Γ . For the objects Pθ and Yθ of Section 2 in case (i) and of Section 3 in case(iii), we claim that

Pθ = Zp⊗Zp[∆ ] P and Yθ = Zp⊗Zp[∆ ]Y,

with P and Y as in 4.2.5. This claim is immediate for Fq(t) as ∆ is trivial, and it is not hard to see for Y incase (i). However, the claim for P is not evident in case (i), so we prove it.

Proof of the claim. Note that R = Zp[θ ]JΓ K and T = Zp[θ ]JΓ K], and note that TR = T(2)R of this section

is T[θ ]JΓ K, where T is as in 2.2.1. The claim for P follows if we can show that the map T (n) 7→ T (n) onHecke operators induces an isomorphism

Tθ/Iθ

∼−→ Zp⊗Zp[∆ ] (TR/IT ),

where I is the Eisenstein ideal of 2.2.1.For a prime ` not dividing N p, the action of Fr−1

` on V is multiplication by θ(`)[`]−1, from which itfollows that a(`k) = θ(`)k[`]−k if ` - N p. On the other hand, condition (3) forces a(`k) = 0 for all k ≥ 1for primes ` dividing N p. The algebra TR contains diamond operators 〈a〉 for a ∈ Γ . This follows from theidentity 〈`〉= `−1(T (`)2−T (`2)) for ` - N p, which also allows us to compute that 〈`〉 ≡ θ(`)[`]−1 mod IT .Thus, IT is generated by T (`)−1−`〈`〉 and 〈`〉−θ(`)[`]−1 for primes ` -N p and T (`)−1 for primes ` |N p.

Noting that the image in TR/IT of every group element is also the image of an element of T[θ ], we nowsee that the map T (n) 7→ T (n) induces an isomorphism (T/I)[θ ] ∼−→ TR/IT of Z[∆ ]-modules, where a ∈ ∆

acts by θ(a)〈a〉−1 on the left and θ(a)〈a〉−1 ≡ [a] mod IT on the right. The induced map on ∆ -coinvariantsis the desired isomorphism. ut


4.2.11. In setting (Ad), we have considered Galois cohomology groups of families of (d−1)-dimensionalGalois representations in the variables given by Iwasawa theory. In case (i) of (Ad), for instance, V is afamily of Galois representations in the cyclotomic variable. Of course, there are other families of Galoisrepresentations, such as Hida families, and we would like to consider them. Therefore, we introduce twoadditional settings (B3) and (Cd) of study. We do not exclude any representations that are new at N fromour families. Perhaps we should, but we prefer a simpler presentation.

4.2.12. We describe setting (B3), in which we work in case (i) for d = 3.Let h and T be as in Section 2.4, and consider the pair (h,T(−1)), where denotes the new-at-N

part. Condition (1) holds for this pair (see 2.4.2). As a consequence of Poincaré duality, the ordinary étalehomology group T may be identified with the Tate twist of the ordinary étale cohomology group as h[GF ]-modules. The characteristic polynomials of Fr` and T (`)∈ h agree on the cohomology T(−1) for any prime` 6= p. Thus, condition (2) is satisfied as well, and the map φT(−1) may be taken to be the identity map onHecke operators.

Similarly to setting (Ad), we consider R = hJΓ K and T = hJΓ K]⊗h T(−1). The conditions (1)–(3) are

again satisfied for (R,T ), and we see that we have φT as in (3) such that a(n) = T (n)[n]−1 for n prime to p.The Eisenstein ideal IT of TR is then generated by

1⊗T (n)− ∑m|n

(m,p)=1

mT (m)[m]−1⊗1 ∈ lim←−r

hJΓ K⊗T(3)(N pr)

for all n≥ 1. Note also that we have an R-module isomorphism

Y = H2ét(Z[ 1

p ],T (3))∼= lim←−

rH2

ét(Z[ζpr , 1p ]

+,T(2)).

4.2.13. We describe setting (Cd), in which we work in case (iii) for d ≥ 2.Let us denote by Y (d−1)

1 (N) the Drinfeld modular variety of dimension d−2 for K(d−1)1 (N) over F . We

define T byT = image(Hd−2

c,ét (Y(d−1)1 (N)/F ,Zp)

→ Hd−2ét (Y (d−1)

1 (N)/F ,Zp)),

where denotes the new part (in an appropriate sense). We then let R be the Zp-submodule of EndZp(T )generated by the Hecke operators T (n) for nonzero ideals n of O.

We imagine but, for d ≥ 4, are not certain that conditions (1)–(3) hold in this case and that we have φTsuch that a(n) = T (n) for all n. In any case, we may define the Eisenstein ideal IT of TR to be generated by

1⊗T (n)−∑d|n

N(d)T (d)⊗1 ∈ R⊗T(d)(N),

for the nonzero ideals n of O. This Eisenstein ideal is all that we need to consider our question.

4.2.14. Our question for (B3) and (Cd) is the same as it was for (Ad), so we can ask it for all:

Question 4. Is there a canonical isomorphism ϖ : P ∼−→Y of R-modules in any of the settings (Ad), (B3), or(Cd)?

This question, which has been formulated rather carelessly, is still not fine enough to be a conjecture.We have more questions than answers: for instance, are the hypotheses that we have made sufficient, andto what extent are they necessary? What happens for the prime p = 2? We do not wish to exclude it fromconsideration. We have made many subtle choices that influence the story in profound yet inapparent ways:e.g., of congruence subgroups, Hecke algebras, Eisenstein ideals, and étale cohomology groups. Have wemade the right choices for a correspondence? We are glad if the reader is inspired to answer these questions.

4.2.15. We end with our hope that it is possible to explicitly define the maps ϖ that are the desired isomor-phisms in the settings (Ad), (B3), and (Cd).

The groups S(N) often have explicit presentations very similar to those of Section 2.1. These are foundin the work of Cremona [Cr], Ash [As], Kondo-Yasuda [KY], and others. So, explicit definitions of ϖ andaffirmative answers to our questions would give explicit presentations of the arithmetic object Y .

The map ϖ should take a modular symbol to a cup product of d special units. As explained above, thishas been done in cases (i) and (iii) for d = 2. Beyond these, the settings in which we hope to do this are:


• (A2) in case (ii), using cup products of two elliptic units,• (B3) using cup products of three Siegel units,• (Cd) using cup products of d of the Siegel units in [KY].

Goncharov has made closely related investigations into the first two of these settings [Go].

Acknowledgements The work of the first two authors (resp., third author) was supported in part by the National ScienceFoundation under Grant Nos. DMS-1001729 (resp., DMS-0901526).

References

As. Ash, A.: Cohomology of congruence subgroups of SLn(Z). Math. Ann. 249, 55–73 (1980)Bu. Busuioc, C.: The Steinberg symbol and special values of L-functions. Trans. Amer. Math. Soc. 360, 5999–6015 (2008)Ca. Carlitz, L.: A class of polynomials. Trans. Amer. Math. Soc. 43, 167–182 (1938)Cr. Cremona, J.: Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields. Compositio

Math. 51, 275–324 (1984)Dr1. Drinfeld, V.: Parabolic points and zeta functions of modular curves. Funkcional. Anal. i Priložen. 7, 83–84 (1973)Dr2. Drinfeld, V.: Elliptic modules. Mat. Sb. 94, 594–627, 656 (1974)Em. Emerton, E.: The Eisenstein ideal in Hida’s ordinary Hecke algebra. Int. Math. Res. Not. IMRN15, 793–802 (1999)Fu. Fukaya, T.: Coleman power series for K2 and p-adic zeta functions of modular forms. In: Kazuya Kato’s fiftieth

birthday. Doc. Math. Extra Vol., 387–442 (2003)FK. Fukaya, T., Kato, K.: On conjectures of Sharifi. Preprint, 121 pagesGe. Gekeler, E.: A note on the finiteness of certain cuspidal divisor class groups. Israel J. Math. 118, 357–368 (2000)Go. Goncharov, A.: Euler complexes and geometry of modular varieties. Geom. Funct. Anal. 17, 1872–1914 (2008)Ha. Hayes, D.: Explicit class field theory for rational function fields. Trans. Amer. Math. Soc. 189, 77–91 (1974)Hi. Hida, H.: Iwasawa modules attached to congruences of cusp forms. Ann. Sci. École Norm. Sup. 19, 231–273 (1986)HP. Harder, G., Pink, R.: Modular konstruierte unverzweigte abelsche p-Erweiterungen von Q(ζp) und die Struktur ihrer

Galoisgruppen. Math. Nachr. 159, 83–99 (1992)Ka. Kato, K.: p-adic Hodge theory and values of zeta functions of modular forms. In: Cohomologies p-adiques et applica-

tions arithmétique, III. Astérisque, vol. 295, pp. 117–290. Soc. Math. France, Paris (2004)Ko. Kondo, S.: Euler systems on Drinfeld modular curves and zeta values. Dissertation, Univ. of Tokyo (2002)KY. Kondo, S.,Yasuda, S.: Zeta elements in the K-theory of Drinfeld modular varieties. Math. Ann. 354, 529–587 (2012)Ku. Kurihara, M.: Ideal class groups of cyclotomic fields and modular forms of level 1. J. Number Theory 45, 281–294

(1993)Ma. Manin, J.: Parabolic points and zeta-functions of modular curves. Math. USSR Izvsetija 6, 19–64 (1972)MW. Mazur, B., Wiles, A.: Class fields of abelian extensions of Q. Invent. Math. 76, 179–330 (1984)McS. McCallum, W., Sharifi, R.: A cup product in the Galois cohomology of number fields. Duke Math. J. 120, 269–310

(2003)Oc. Ochiai, T.: On the two-variable Iwasawa main conjecture. Compos. Math. 142, 1157–1200 (2006)Oh1. Ohta, M.: On the p-adic Eichler-Shimura isomorphism for Λ -adic cusp forms. J. reine angew. Math. 463, 49–98 (1995)Oh2. Ohta, M.: Ordinary p-adic étale cohomology groups attached to towers of elliptic modular curves. II. Math. Ann. 318,

557–583 (2000)Oh3. Ohta, M.: Congruence modules related to Eisenstein series. Ann. Éc. Norm. Sup. 36, 225–269 (2003)Pa. Pal, A.: The rigid analytical regulator and K2 of Drinfeld modular curves. Publ. Res. Inst. Math. Sci. 46, 289–334

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Weber’s Class Number One Problem

Takashi Fukuda, Keiichi Komatsu and Takayuki Morisawa

Abstract Let p and ` be prime numbers. In this paper, we consider the `-indivisibility of the class numbersof the intermediate fields in the cyclotomic Zp-extension of Q. Moreover, we study the class numbers of theintermediate fields in the composite of such extensions.

1 Weber’s class number one problem

Two hundred years ago, Gauss made the following conjecture.

Conjecture 1. There are infinitely many real quadratic fields with class number one.

This conjecture is still open not only for quadratic fields but also for algebraic number fields with arbitraryfinite degree over Q.

Conjecture 2. There are infinitely many algebraic number fields with class number one.

To study this problem, we focus our attention on the cyclotomic Zp-extension of the rational number fieldQ.

Let p be a prime number. We denote by B(p∞) and B(pn) the cyclotomic Zp-extension of Q andits n-th layer, respectively. Then, for example, we know that B(2n) = Q(cos(2π/2n+2)) and B(3n) =Q(cos(2π/3n+1)). We denote by h(pn) the class number of B(pn). Now, we consider the following problem.

Problem 1 (Weber’s Class Number Problem). Is the class number h(pn) equal to one for any positiveinteger n?

Weber proved that h(21) = h(22) = h(23) = 1. Later, it was shown that h(24) = 1 by Bauer [2], Cohn [4]and Masley [19] and h(25) = 1 by Linden [18]. In [19] and [18], we know that h(31) = h(32) = h(33) =h(51) = h(71) = 1. Linden also showed that h(26) = h(34) = h(52) = h(111) = h(131) = 1 under the gen-eralized Riemann hypothesis.

However, it is very hard to compute the whole class number. So we focus on the `-indivisibility of h(pn).

Problem 2. Is the class number h(pn) coprime to a prime number ` for any non-negative integer n?

Takashi FukudaDepartment of Mathematics, College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba 275-0005,Japan, e-mail: [email protected],

Keiichi KomatsuDepartment of Mathematics, School of Fundamental Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku,Tokyo 169-8555, Japan, e-mail: [email protected]

Takayuki MorisawaDepartment of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku, Yokohama, Kana-gawa, 223-8522, Japan, e-mail: [email protected]

161

[email protected]

[email protected]

[email protected]

162 Takashi Fukuda, Keiichi Komatsu and Takayuki Morisawa

In the case ` = p, it was shown that p does not divide h(pn) by Weber [28] (p = 2) and Iwasawa [17](in general). Thus, we study the non-p-part of h(pn). In this case, that is, ` 6= p, there is Washington’s result[26] which says that the `-part of h(pn) is bounded as n tends to ∞. However, we can not get any informationon indivisibility from it. On the `-indivisibility, there is an approach of Horie [8, 9, 10, 11] and Horie-Horie[12, 13, 14, 15] which tries to attack h(pn) by using the cyclotomic units. The following theorem is a partof their results.

Theorem 1 (Horie, Horie-Horie).

(1) Let p= 2. If a prime number ` satisfies ` 6≡ ±1 (mod 8), then h(2n) is coprime to ` for any non-negativeinteger n.

(2) Let p be a prime number with 3≤ p≤ 23. If a prime number ` is a primitive root modulo p2, then h(pn)is coprime to ` for any non-negative integer n.

Although these results were very striking and very effective, there were many small prime numbers pand ` for which we did not know whether ` divides h(pn) or not. For example, it was not known whether` divides h(2n) for ` = 7, 17, 23, 31, . . . . In order to consider the `-indivisibility of h(pn) for such primenumbers p and `, we showed the following theorem.

Theorem 2. (1) Let p = 2, ` an odd prime number and 2s the exact power of 2 dividing `− 1 or `2− 1according as `≡ 1 (mod 4) or not. Put

m(2, `) = 2s+⌊

12

log2(`−1)⌋−2,

where bxc is the greatest integer not exceeding a real number x. If ` does not divide h(2m(2,`)), thenh(2n) is coprime to ` for any non-negative integer n.

(2) Let p = 3, ` a prime number different from 3 and 3s the exact power of 3 dividing `2−1. Put

m(3, `) = 2s+⌊

12

log3(`−1)+12

⌋−1.

If ` does not divide h(3m(3,`)), then h(3n) is coprime to ` for any non-negative integer n.

We prove the above theorem by using the method of Sinnott and Washington (see [27] section 16) andthe Iwasawa main conjecture proved by Mazur-Wiles [20]. Theorem 2, together with numerical calculationsbased on [5] and [21], allows us to obtain the following theorem.

Theorem 3 (see [5, 6, 7] and [21, 22]). If ` is a prime number with ` < 109, then h(2n) and h(3n) arecoprime to ` for any non-negative integer n.

We also know the following theorem.

Theorem 4 (see [22], [24], [25]). Let p and ` be different prime numbers. We denote by f the inertia degreeat ` in Q(µ2p) over Q, where µm is the group of all m-th roots of unity. Let ps be the exact power of p dividing` f −1 and c = (p−1)ps−1. We put

B(p,s, f ) =

(Ac · c!)1/ f , if p = 2(√

2c · c!

)1/ f, if p = 3((√

6p2

)c· c!)1/ f

, if p≥ 5

where A is the constant defined by A = 0.80785 · · · . Then h(pn) is coprime to ` for any non-negative integern, if ` > B(p,s, f ).

By combining Theorem 3 and Theorem 4, we get the following corollary.

Corollary 1. (1) If ` is a prime number with ` 6≡ ±1 (mod 32), then h(2n) is coprime to ` for any non-negative integer n.

(2) If ` is a prime number with ` 6≡ ±1 (mod 27), then h(3n) is coprime to ` for any non-negative integern.

Weber’s Class Number One Problem 163

2 Composites of Zp-extensions of Q

In this section, we consider the class number of the intermediate field in the Z-extension of Q.We denote by B(∞) the composite of B(p∞) for all prime numbers p. Then the Galois group of B(∞) over

Q is isomorphic to Z as a topological group. For any positive integer N ∈ N, we denote by B(N) and h(N)the unique subfield of B(∞) with degree N over Q and its class number, respectively. On this extension,Coates [3] conjectured the following.

Conjecture 3. There exists a number CQ not depending on N, such that h(N)≤CQ for all N ∈ N.

In this direction, Coates-Liang-Mihailescu varified that h(N) = 1 for 1≤ N ≤ 28 (see [3]).Originally, we wanted to attack the above conjecture. However, the Z-extension is too large to study.

Therefore we restrict to the case of ZΣ -extensions.Let Σ be a non-empty finite set of prime numbers. Put ZΣ = ∏p∈Σ Zp and

NΣ =

∏p∈Σ

pnp

∣∣∣∣∣ np ∈ Z≥0

.

Then the family B(N) | N ∈ NΣ is the family of all intermediate fields of ZΣ -extension of Q with finitedegree over Q. We consider the following problem.

Problem 3. Let Σ be a non-empty finite set of prime numbers and ` a prime number. Is the class numberh(N) coprime to ` for all N ∈ NΣ or not?

Let ΩΣ = Q(µN | N ∈NΣ ) and denote by ΩΣ (`) the decomposition field of a prime number ` in ΩΣ overQ. For an intermediate field F of ΩΣ with finite degree over Q, we denote by cond(F) the conductor of Fand put d = ϕ(cond(F)) and f = d[F : Q]−1 where ϕ is the Euler function and [F : Q] is the degree of Fover Q. We also put

D(Σ ,F) = ` : prime number 6∈ Σ |ΩΣ (`) = F

and

B(Σ ,F) =

(∏p∈Σ

2p

)d

·d!

1/ f

.

Then, on the `-indivisibility, we have the following theorem.

Theorem 5 (see [23]). If a prime number ` ∈ D(Σ ,F) satisfies ` > B(Σ ,F), then h(N) is coprime to ` forall N ∈ NΣ .

The above theorem says that the class numbers are coprime to ` if ` is sufficiently large in comparison withthe conductor of its decomposition field in ΩΣ .

Example 1. Put Σ = 3,5 and F = Q(√−15). Then we have cond(F) = 15, d = 8 and f = 4. We obtain

B(3,5,Q(√−15)) = 51013.2075 · · · .

For example, `= 51197, 51203, 51287, 51347 and 51383 are prime numbers and their decomposition fieldis Q(

√−15). Therefore, Theorem 5 implies that h(3n5m) is coprime to ` for all n and m.

On the other hand, there are small prime divisors of h(N). The following prime divisors are all we know.

Theorem 6. (1) We have 31 | h(2 · 31), 1546463 | h(2 · 1546463), 73 | h(3 · 73), 18433 | h(28 · 18433),114689 | h(210 ·114689), 73 | h(3 ·73), 487 | h(34 ·487), 238627 | h(34 ·238627) and 2251 | h(52 ·2251).

(2) We have 107 | h(2 ·53).

The case (1) is a consequence of Proposition 2 in [16] and the case (2) was found by using Theorem 2 in[1].

Acknowledgements The authors thank the organizers of Iwasawa 2012 for giving us the opportunity to talk. The authorsalso thank the referee for reading this paper carefully and offering several invaluable suggestions.

164 Takashi Fukuda, Keiichi Komatsu and Takayuki Morisawa

References

1. Aoki M. and Fukuda T.: An algorithm for computing p-class groups of abelian number fields, Lecture Notes in ComputerScience, 4076 (2006), 56–71.

2. Bauer H.: Numerische Bestimmung von Klassenzahlen reeller zyklischer Zahlkörper, J. Number Th., 1 (1969), pp. 161–162.

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4. H. Cohn, A numerical study of Weber’s real class number calculation I, Numer. Math. 2, 2 (1960), 347–362.5. T. Fukuda and K. Komatsu, Weber’s class number problem in the cyclotomic Z2-extension of Q, Experiment. Math. 18-2

(2009), 213–222.6. T. Fukuda and K. Komatsu, Weber’s class number problem in the cyclotomic Z2-extension of Q, II, J. Théor. Nombres

Bordeaux 22-2 (2010), 359–368.7. T. Fukuda and K. Komatsu, Weber’s class number problem in the cyclotomic Z2-extension of Q, III, Int. J. Number Theory

7-6 (2011), 1627–1635.8. K. Horie, Ideal class groups of Iwasawa-theoretical abelian extensions over the rational field, J. London Math. Soc. 66

(2002), 257–275.9. K. Horie, Primary components of the ideal class group of the Zp-extension over Q for Typical Inert Primes, Proc. Japan

Acad. Ser. A Math. Sci., 81 (2005), no.3, 40–43.10. K. Horie, The ideal class group of the basic Zp-extension over an imaginary quadratic field, Tohoku Math. J. (2), 57

(2005), 375–394.11. K. Horie, Certain primary components of the ideal class group of the Zp-extension over the rationals, Tohoku Math. J.,

59 (2007), 259–291.12. K. Horie and M. Horie, The narrow class groups of some Zp-extensions over the rationals, Acta Arith., 135 (2008), no.2,

159–180.13. K. Horie and M. Horie, The ideal class group of the Zp-extension over the rationals, Tohoku Math. J., 61 (2009), 551–570.14. K. Horie and M. Horie, The narrow class groups of the Z17- and Z19-extensions over the rational field, Abh. Math. Semin.

Univ. Hambg. 80-1 (2010), 47–57.15. K. Horie and M. Horie, The ideal class group of the Z23-extension over the rational field, Proc. Japan Acad., 85 (2009),

Ser. A, 155–159.16. Inatomi A.: On Zp-extensions of real abelian fields, Kodai Math. J., 12 (1986), 420–422.17. K. Iwasawa, A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg 20 (1956), 257–258.18. F. J. van der Linden, Class number computations of real abelian number fields, Math. Comp. 39 (1982), 693–707.19. J. M. Masley, Class numbers of real cyclic number fields with small conductor, Compositio math. 37 (1978), 297–319.20. B. Mazur and A. Wiles, Class fields of abelian extensions of Q, Invent. Math. 76 (1984), 179-330.21. T. Morisawa, A class number problem in the cyclotomic Z3-extension of Q, Tokyo J. Math. 32 (2009), 549–558.22. T. Morisawa, Mahler measure of the Horie unit and Weber’s class number problem in the cyclotomic Z3-extension of Q,

Acta Arith. 153-1 (2012), 35–49.23. T. Morisawa, On the `-part of the Zp1 ×·· ·×Zps -extension of Q, J. Number Theory 133 no.6 (2013), 1814–1826.24. T. Morisawa and R. Okazaki, Mahler measure and Weber’s class number problem in the cyclotomic Zp-extension of Q

for odd prime number p Tohoku Math. J. (2), 65 no.2, 253–272.25. R. Okazaki, On a lower bound for relative units, Schinzel’s lower bound and Weber’s class number problem, preprint.26. L. C. Washington, The non-p-part of the class number in a cyclotomic Zp-extension, Invent. Math. 49 (1978), 87–97.27. L. C. Washington, Introduction to cyclotomic fields, 2nd edition, Graduate Texts in Math., 83 (1997), Springer-Verlag,

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On p-adic Artin L-functions II

Ralph Greenberg

1 Introduction

Let p be a prime. Iwasawa’s famous conjecture relating Kubota-Leopoldt p-adic L-functions to the structureof certain Galois groups has been proven by Mazur and Wiles in [11]. Wiles later proved a far-reachinggeneralization involving p-adic L-functions for Hecke characters of finite order for a totally real numberfield in [16]. As we discussed in [5], an analogue of Iwasawa’s conjecture for p-adic Artin L-functionscan then be deduced. The formulation again involves certain Galois groups. However, one can reformulatethis result in terms of Selmer groups for the Artin representations. There are several advantages to such areformulation. First of all, it fits perfectly into the much broader framework described in [6] which relatesthe p-adic L-function for a motive to the corresponding Selmer group. The crucial assumption in [6] thatthe motive be ordinary at p (or at least potentially ordinary) is satisfied by an Artin motive and all of its Tatetwists.

A second advantage of a reformulation involving Selmer groups is that the issue of how to define the µ-invariant becomes resolved in a natural and transparent way. Thirdly, the arguments in [5] can be simplified.In particular, there is no need for singling out the class of Artin representations which are called type S in[5]. The purpose of this paper is to explain these advantages.

Suppose that F is a totally real number field. Consider an Artin representation

ρ : GF −→ AutE(V ) ,

where GF is the absolute Galois group of F and V is a finite dimensional vector space over a finite extensionE of Qp. We will assume that ρ is totally even. This means that ρ factors through ∆ = Gal(K/F), where Kis a finite extension of F which is also totally real. Let O be the ring of integers of E. Let T be an O-latticein V which is GF -invariant. Furthermore, let D =V/T , a discrete O-module.

Let F∞ denote the cyclotomic Zp-extension of F . The Selmer group associated to D over F∞ is definedby

SelD(F∞) = ker(

H1(F∞,D)−→∏η -p

H1(F∞,η ,D))

.

Here η runs over all the primes of F∞ except for the finitely many primes lying over p. The archimedeanprimes are included in the product, although this is only important when p = 2. One defines the field F∞,η

to be the union of the η-adic completions of the finite extensions of F contained in F∞.To relate the above definition to the way Selmer groups are defined in [6], note that if ρ is a totally even

Artin representation of GF over C, then the Artin L-function L(s,ρ) does not have a critical value at s = 1in the sense of Deligne. However, its value at s = 1−n is critical in that sense when n is even and positive.One can write

L(1−n,ρ) = L(1,ρ(n))

where ρ(n) is the n-th Tate twist. The underlying representation space for ρ(n) over E is V (n) = V ⊗ χnF ,

where χF : GF → Z×p is the p-power cyclotomic character. In the notation of [6], we have F+V (n) =V (n)when n ≥ 1. (This is so for all the primes above p.) Let T (n) = T ⊗ χn and D(n) = V (n)/T (n). The

165

166 Ralph Greenberg

corresponding Selmer group SelD(n)(F∞), as it is defined in [6], is just as above, but with D(n) replacing D.Let d = [F(µq) : F ], where q = p when p is odd and q = 4 when p = 2. If we take n ≡ 0 (mod d), thenD(n) ∼= D for the action of GF∞

. Thus, the two Selmer groups are then the same, although the action ofGal(F∞/F) on those groups is somewhat different. (See remark 6.)

Since SelD(F∞) is a discrete O-module and ΓF = Gal(F∞/F) acts naturally and continuously on it, wecan regard SelD(F∞) as a discrete Λ(O,F)-module, where Λ(O,F) =O[[ΓF ]]. It is not difficult to show that thePontryagin dual XD(F∞) of SelD(F∞) is a finitely generated, torsion Λ(O,F)-module. (See proposition 1) Wedenote the characteristic ideal of that Λ(O,F)-module by Iρ . It is a principal ideal in the ring Λ(O,F). As thenotation suggests, this ideal depends only on ρ , and not on the choice of the Galois-invariant O-lattice T , aswe show in proposition 2.

Another discrete Λ(O,F)-module to be considered is H0(F∞,D). Its Pontryagin dual YD(F∞) is clearly afinitely-generated O-module and hence a torsion Λ(O,F)-module. Let Jρ denote the characteristic ideal ofYD(F∞). This ideal is nontrivial if and only if ρ has at least one irreducible constituent which factors throughΓF .

The p-adic L-function associated to ρ will be denoted by Lp(s,ρ). It is characterized by a certain inter-polation property. In case ρ is 1-dimensional, these functions have been constructed by Deligne and Ribetin [8], by Cassou-Noguès in [3], and by Barsky in [2]. One can then define Lp(s,ρ) if ρ has arbitrarydimension by using a classical theorem of Brauer from group theory.

One can associate to Lp(s,ρ) a certain element θρ in the fraction field of Λ(O,F). For an odd prime p, theMain Conjecture is the assertion that the fractional ideals Λ(O,F)θρ and Iρ J−1

ρ are the same. This is provedin section 4 as a consequence of theorems of Wiles proved in [16]. For p = 2, there is an extra power of 2in the formulation, but this case appears to still be open.

2 Basic results concerning the Selmer group

We will prove several useful propositions. We continue to make the same assumptions as in the introduction.In particular, F is a totally real number field and ρ is a totally even Artin representation of GF defined overa field E, a finite extension of Qp. The ring of integers in E is denoted by O. Let m be the maximal ideal ofO.

We will use the traditional terminology for modules over a topological ring Λ . If S is a discrete Λ -module,and X is its Pontryagin dual, then we say that S is a cofinitely-generated Λ -module if X is finitely-generated.If X is a torsion Λ -module, we say that S is cotorsion.

Suppose that V , T , and D =V/T are as in the introduction. Let d = dimE(V ). As an O-module, we haveD∼= (E/O)d . The Selmer group SelD(F∞) is a discrete Λ(O,F)-module.

Proposition 1. The Λ(O,F)-module SD(F∞) is cofinitely-generated and cotorsion.

Proof. Suppose that ρ factor through Gal(K/F), where K is a totally real, finite Galois extension of F . Let∆ =Gal(K∞/F∞) and let M∞ be the maximal abelian pro-p extension of K∞ which is unramified at the primesof K∞ not lying over p (including the archimedean primes). One can consider X(K∞) = Gal(M∞/K∞) as amodule over Λ(Zp,K) = Zp[[ΓK ]]. A well-known theorem of Iwasawa asserts that X(K∞) is finitely-generatedand torsion as a Λ(Zp,K)-module. The fact that it is torsion is equivalent to the fact that the so-called weakLeopoldt conjecture is valid for K∞/K.

We have H0(K∞,D) = D. Also, H1(∆ ,D) is finite. Hence the restriction map

H1(F∞,D)−→ H1(K∞,D)∆ (1)

has finite kernel. We can identify ΓK with Gal(F∞/K ∩F∞), a subgroup of ΓF . The map (1) is then ΓK-equivariant. Now H1(K∞,D) = Hom(Gal(Kab

∞ /K∞),D), where Kab∞ is the maximal abelian extension of K∞.

Since the inertia subgroups of Gal(Kab∞ /K∞) for all primes η - p generate Gal(Kab

∞ /M∞), it is clear that the

On p-adic Artin L-functions II 167

image of SelD(F∞) under the map (1) is contained in Hom(X(K∞),D), which is a cofinitely-generated, co-torsion Λ(Zp,K)-module according to Iwasawa’s theorem. Since (1) has finite kernel, it follows that SelD(F∞)is cofinitely-generated and cotorsion as a Λ(Zp,K)-module, and therefore as a Λ(O,F)-module. ut

Remark 1. With the notation of the above proof, the cokernel of the map (1) is also finite. This followsfrom the fact that H2(∆ ,D) is finite. Assume that the order of im(ρ) is not divisible by p. We can thenassume that p - |∆ |. In particular, K ∩F∞ = F . Hence the map ΓK → ΓF is an isomorphism. We then haveGal(K∞/F)∼= ∆ ×ΓF . Furthermore, (1) is an isomorphism. The induced map

SelD(F∞)−→ Hom∆ (X(K∞),D) (2)

is also easily verified to be an isomorphism. In addition to assuming p - |∆ |, assume that ρ is absolutelyirreducible. Let eρ be the idempotent for ρ in O[∆ ]. Then

Hom∆ (X(K∞), D) ∼= HomO[∆ ]

(X(K∞)⊗Zp

O, D) ∼= HomO[∆ ]

(eρ X(K∞)⊗Zp

O, D).

Thus, the Λ(O,F)-modules XD(F∞) and eρ X(K∞)⊗Zp O are closely related. In fact, the characteristic idealof the second module is Id

ρ .

Remark 2. Suppose that X is any Λ(O,F)-module and that I is the characteristic ideal of X . Let π be agenerator of m, the maximal ideal of O. The µ-invariant of X will be denoted by µF(X). It is the integerµ characterized by I ⊆ πµΛ(O,F), I 6⊆ πµ+1Λ(O,F). A conjecture of Iwasawa (at least for odd primes p)asserts that the µ-invariant of the Λ(O,K)-module XK∞

should vanish. This should be true even for p = 2. Ifthis is so, then the proof of proposition 1 would show that µF

(XD(F∞)

)= 0.

It will be useful to have an alternative definition of SelD(F∞). Let Σ be a finite set of primes of Fcontaining the archimedean primes, the primes lying over p, and the ramified primes for ρ . For each v ∈ Σ ,define

H1v(F∞,D) = lim−→

n

⊕ν |v

H1(Fn,ν ,D)

where, for each n, ν runs over the primes of Fn lying over v. The maps defining the direct limit are inducedby the local restriction maps. If v is a finite prime, then

H1v(F∞,D) =

⊕ν |v

H1(F∞,ν ,D)

where ν runs over the finite set of primes of F∞ lying over v. The p-cohomological dimension of GF∞,ν is 1,and so H2(F∞,ν ,D[π]) = 0. It follows that H1(F∞,ν ,D) is O-divisible. Now assume that v - p. Then, accordingto proposition 2 in [6], the O-corank of H1(F∞,ν ,D) is finite. It therefore follows that the Pontryagin dual ofH1

v(F∞,D) is a torsion-free O-module of finite rank. Hence it is a free O-module. It is also a Λ(O,F)-module.Since its O-rank is finite, it must be a torsion Λ(O,F)-module whose µ-invariant vanishes.

If p is odd and v is an archimedean prime, then H1v(F∞,D) = 0. However, if p = 2, then H1

v(F∞,D) isnontrivial. More precisely, since all the Fn’s are totally real, and ρ is totally even (so that GFn,ν acts triviallyon D), we have

H1(Fn,ν ,D) = H1(R,D) = D[2]∼= (O/2O)d

if ν is archimedean. One sees that H1v(F∞,D) is a direct limit of modules isomorphic to (O/2O)[Gal(Fn/F)]d .

The Pontryagin dual of H1v(F∞,D) is isomorphic to

(Λ(O,F)

/2Λ(O,F)

)d as a Λ(O,F)-module. It is a torsionΛ(O,F)-module, but its µ-invariant is positive and is determined by d = dimE(V ).

One can similarly define H2v(F∞,D) as a direct limit by replacing the H1’s by H2’s. However, for any

finite prime v, the p-cohomological dimension of GF∞,ν is 1 and so H2v(F∞,D) vanishes. This is also true if

v|∞ because H2(R,D) = 0.The following definition is equivalent to the one given in the introduction:

SelD(F∞) = ker(

H1(FΣ/F∞,D)−→ ∏v∈Σ ,v-p

H1v(F∞,D)

). (3)

168 Ralph Greenberg

To verify the equivalence, we first point out that if v is a non-archimedean prime of F and v - p, and if ν is aprime of F∞ lying over v, then F∞,ν is the unramified Zp-extension of Fv. It follows that the restriction mapH1(F∞,ν ,D)→ H1(Funr

v ,D) is injective. Here Funrv is the maximal unramified extension of Fv and contains

F∞,ν . Therefore, requiring a cocycle class to be trivial in H1(F∞,ν ,D) is equivalent to requiring it to be trivialin H1(Funr

ν ,D).Suppose now that φ is a 1-cocycle for GF∞

with values in D. Note that we have H1(FΣ ,D)=Hom(GFΣ,D).

Also, GFΣis generated topologically by the inertia subgroups Iη of GF for all primes η of FΣ lying over

some v 6∈ Σ . Thus, the class [φ ] in H1(F∞,D) has a trivial restriction to all those Iη ’s if and only if [φ ] is in

ker(H1(F∞,D)→ H1(FΣ ,D)

)= im

(H1(FΣ/F∞,D)→ H1(F∞,D)

).

Thus, the cocycle classes in H1(FΣ/F∞,D) can be identified under the inflation map with the cocyle classesin H1(F∞,D) which are unramified at all primes of F∞ not lying over primes in Σ . The equivalence of (3)and the earlier definition follows.

It will also be useful to point out that the global-to-local map in (3) is surjective. This follows proposition2.1 in [8]. It is only proved there for F = Q and odd p, but the argument works if F is totally real and forany p. One assumption is that SelD(F∞) is ΛO,F -cotorsion, which is satisfied by proposition 1 above. Theother assumption is that H0(F∞,D∗) is finite. Here D∗ = Hom(T,µp∞) and the finiteness is clear since F∞ istotally real and H0(R,µp∞) is finite (and even trivial if p is odd).

Let T ′ be another GF -invariant O-lattice in V . Let D′ = V/T ′. We consider XD(F∞) and XD′(F∞) asΛ(O,F)-modules.

Proposition 2. The Λ(O,F)-modules XD(F∞) and XD′(F∞) have the same characteristic ideal.

Proof. As above, let π be a generator of the maximal ideal of O. Scaling by a power of π , we may assumethat T ⊆ T ′. We then have a GF -equivariant map ϕ : D→ D′ with finite kernel. Such a map ϕ is called aGF -isogeny. It is surjective. Let Φ = ker(ϕ). Then Φ ⊆ D[π t ] for some t ≥ 0. There is also a GF -isogenyψ : D′→ D such that ψ ϕ is the map D→ D given by multiplication by π t .

The map ϕ induces a map from SelD(F∞) to SelD′(F∞) whose kernel is killed by π t . Similarly, ψ inducessuch a map from SelD′(F∞) to SelD(F∞) and the compositum is multiplication by π t . It follows that thecharacteristic ideals I and I′ of XD(F∞) and XD′(F∞), respectively, are related as follows: I′ = πsI for somes ∈ Z. Thus, the proposition is equivalent to showing that the µ-invariants for the two modules are equal,and so s = 0.

Assume first that p is odd. In the definition (3), H1v(F∞,D) = 0 when v|∞ and H1

v(F∞,D) has finite O-corank when v ∈ Σ ,v - p. Thus, the µ-invariants for the Pontryagin duals of SelD(F∞) and H1(FΣ/F,D)are equal. The same statement is true for SelD′(F∞) and H1(FΣ/F,D′). And so it suffices to prove thatthe Pontryagin duals of H1(FΣ/F,D) and H1(FΣ/F,D′) have the same µ-invariants. This is sufficient evenfor p = 2. This follows because the map (3) and the corresponding global-to-local map for D′ are bothsurjective. Furthermore, for any archimedean prime v, the µ-invariants of H1

v(F∞,D) and H1v(F∞,D′) are

equal.Using the notation in the proof of proposition 1, we have an exact sequence

H0(FΣ/F∞,D′)−→ H1(FΣ/F∞,Φ)−→ H1(FΣ/F∞,D)−→ H1(FΣ/F∞,D′)

−→ H2(FΣ/F∞,Φ)−→ H2(FΣ/F∞,D) .

Now the µ-invariant of H0(FΣ/F∞,D′) certainly vanishes. Also, H2(FΣ/F∞,D) = 0. One can verify thisfor odd p by using propositions 3 and 4 in [6]. For H2(FΣ/F,D) is Λ(O,F)-cotorsion by proposition 3,and Λ(O,F)-cofree by proposition 4. For p = 2, H2(FΣ/F,D) is still Λ(O,F)-cotorsion. The analogue ofproposition 4 is that

ker(H2(FΣ/F∞,D)→∏

v|∞H2

v(F∞,D))

is Λ(O,F)-cofree, and hence must vanish. However, since H2v(F∞,D) vanishes, it follows that H2(R,D) = 0.

Consequently, we indeed have H2(FΣ/F∞,D) = 0.


To complete the proof, we must show that H1(FΣ/F∞,Φ) and H2(FΣ/F∞,Φ) have the same µ-invariantsas Λ(O,F)-modules. In the statement of the proposition, we can reduce to the case where Φ is killed by π .Let Λ(O,F) = Λ(O,F)/πΛ(O,F). It then suffices to show that

corankΛ(O,F)

(H1(FΣ/F∞,Φ)

)= corank

Λ(O,F)

(H2(FΣ/F∞,Φ)

).

Here Φ is a representation space for GF over O/(π) and is totally even. The Euler-Poincaré characteristicover F∞, which is the alternating sum of the Λ(O,F)-coranks of H i(FΣ/F∞,Φ) for 0≤ i≤ 2, turns out to be0. The above equality follows.

The above assertion about the Euler-Poincaré characteristic for the Gal(FΣ/F∞)-module Φ is a con-sequence of the fact that if Fn denotes the n-th layer in the Zp-extension F∞/F , then the Euler-Poincarécharacteristic for the finite Gal(FΣ/Fn)-module Φ is equal to 1. (See [NSW08], (8.6.14), page 427.) Con-sidering Φ as a vector space over f = O/(π), this means that the alternating sum of the f-dimensions ofH i(FΣ/Fn,Φ) for 0 ≤ i ≤ 2 is equal to 0. For the argument relating this fact to the above assertion aboutthe Λ(O,F)-coranks, we refer the reader to the proof of Proposition 4.1.1 in [7]. The argument there is fora Galois module E[p]⊗α (which is also a f-vector space), but applies to any such finite Galois module Φ ,the only difference being the values of the Euler-Poincaré characteristics over the Fn’s as n varies. ut

Remark 3. As mentioned in remark 2, µ(XD(F∞)

)should always vanish. The above proof would then show

that XD(F∞) and XD′(F∞) are pseudo-isomorphic as Λ(O,F)-modules. This is also true if µ(XD(F∞)

)= 1. In

contrast, for non-Artin motives, the µ-invariant of the Pontryagin dual of a Selmer group can be nonzeroand can change under isogeny. This phenomenon was first pointed out by Mazur in [10]. The exact changein the µ-invariant under an isogeny is studied in [22] and [PR94]. In fact, proposition 2 is just a special caseof the main theorem in [PR94] when p is an odd prime.

Remark 4. If ϕ : D→D′ is a GF -isogeny and ker(φ) = D[mt ] for some t ≥ 0, then D∼= D′ as GF -modules.This follows because the maximal ideal m of O is a principal ideal. Any other GF -isogeny will be callednontrivial. Such GF -isogenies ϕ exist if and only if T/πT is reducible as a GF -representation space over theresidue field O/(π). Now, if ρ is irreducible over E and im(ρ) has order prime to p, then it is well-knownthat T/πT is also irreducible. In contrast, if im(ρ) has order divisible by p, then T/πT may be reducibleeven if ρ is irreducible.

Proposition 3. Suppose that ρ1 and ρ2 are totally even Artin representations of GF . Let ρ = ρ1⊕ρ2. ThenIρ = Iρ1 Iρ2 and Jρ = Jρ1Jρ2 .

Proof. We assume that E is a field of definition for ρ1,ρ2, and ρ . Let V1 and V2 be the underlyingrepresentations spaces for ρ1 and ρ2 over E, and let V = V1 ⊕V2. Let T1 and T2 be Galois invari-ant O-lattices in V1 and V2. Let T = T1 ⊕ T2. Then D = D1 ⊕D2. With this choice of O-lattices, it isclear that SelD(F∞) ∼= SelD1(F∞)⊕SelD2(F∞). The first equality in the proposition follows. We also haveH0(F∞,D)∼= H0(F∞,D1)⊕H0(F∞,D2), giving the second equality. ut

Suppose that F ′ is a totally real, finite extension of F and that ρ ′ is a totally even Artin representation ofGF ′ . Thus, ρ ′ factors through Gal(K′/F ′), where K′ is totally real. We can then define ρ = IndGF

GF ′(ρ ′). Then

ρ is an Artin representation of GF and factors through Gal(K/F), where K is the Galois closure of K′ over F .Note that K is totally real, and hence ρ is also totally even. Furthermore, there is an injective homomorphismΓF ′ → ΓF . Therefore, we can regard Λ(O,F ′) as a subring of Λ(O,F). One sees easily that Λ(O,F) is a finiteintegral extension of Λ(O,F ′) and that the degree is [ΓF : ΓF ′ ] = [F ′∩F∞ : F ]. The characteristic ideals I andJ are essentially unchanged by induction. To be precise, we have

Proposition 4. With the above notation, we have Iρ = Iρ ′Λ(O,F) and Jρ = Jρ ′Λ(O,F).

To simplify notation in the following proof, we will write IndFF ′(ρ

′) in place of IndGFGF ′

(ρ ′). We will also writeρ∞ and ρ ′∞ for the restrictions of ρ and ρ ′ to F∞ and F ′∞, respectively.

Proof. We consider separately the two cases where F ′∩F∞ = F and F ′ ⊂ F∞. That will suffice because ifE = F ′∩F∞, then E∞ = F∞ and IndF

E(IndE

F ′(ρ′))= ρ .

170 Ralph Greenberg

Suppose first that F ′∩F∞ =F . In this case, we can identify ΓF ′ with ΓF and hence Λ(O,F ′) with Λ(O,F). Forbrevity, let G = GF ,G′ = GF ′ , and N = GF∞

, a normal supgroup of G. Then N ∩G′ = GF ′∞ . Now supposethat K is a finite, totally real Galois extension of F which contains F ′ and such that ρ ′ factors throughGal(K/F ′). Then ρ factors through Gal(K/F). Furthermore, since NG′=G, we have [G : G′] = [N : N∩G′],and it follows that

ρ|N = IndGG′(ρ

′)∣∣N = IndN

N∩G′(ρ′|N) .

Consequently, ρ∞∼= IndF∞

F ′∞(ρ ′∞).

Choose the Galois invariant O-lattices for ρ and ρ ′ so that IndGG′(D

′) = D. Here we can replace G andG′ by N and N ∩G′. Then H0(F∞,D) ∼= H0(F ′∞D′), and the isomorphism is equivariant for the action ofΓF = ΓF ′ . It follows that Jρ = Jρ ′ .

Note that K∞ = KF∞ contains F ′∞. Let M∞ be defined exactly as in the proof of proposition 1. Then, forthe reason given in that proof, we have injective maps

SelD(F∞)→ H1(M∞/F∞,D), SelD′(F′∞)→ H1(M∞/F ′∞,D

′) . (4)

The cokernels of these maps are finite. The proof is the same for both maps, and so we just discuss thefirst map. If η is any nonarchimedean prime of M∞ not dividing p, let Aη denote the corresponding inertiasubgroup of Gal(M∞/F∞). Note that the image of the first map in (4) consists of 1-cocycle classes whichhave a trivial restriction to H1(Aη ,D) for all those η’s. It is enough to consider the Aη ’s for η |ν , where ν

is a prime of F∞, ν - p, and ν is ramified in K∞/F∞. Otherwise, Aη is itself trivial. Only finitely many suchprimes ν exist. It suffices to just consider one η for each ν .

Since M∞/K∞ is unramified at any η - p, it follows that Aη is isomorphic to the corresponding inertiasubgroup of Gal(K∞/F∞) and hence is finite. Therefore, the finiteness of the above cokernel follows fromthe fact that if A is any finite subgroup of Gal(M∞/F∞), then H1(A,D) is finite.

Let U = GM∞. Then we can identify IndG/U

G′/U (ρ′) with ρ , viewed as a representation of G/U =

Gal(M∞/F), and also of the subgroup Gal(M∞/F∞). According to Shapiro’s Lemma, we then have a canon-ical isomorphism

H1(M∞/F∞,D) −→ H1(M∞/F ′∞,D′) (5)

The map is ΓF -equivariant and so the isomorphism is as discrete Λ(O,F)-modules. Their Pontryagin duals areisomorphic as Λ(O,F)-modules. It follows that the Pontryagin duals of SelD(F∞) and SelD′(F ′∞) are pseudo-isomorphic and therefore that Iρ = Iρ ′ , as stated. We remark in passing that, with a little more care, one canverify that (5) actually defines an isomorphism of SelD(F∞) to SelD′(F ′∞).

Suppose now that F ′ ⊂ F∞. Then F ′∞ = F∞ and ΓF ′ is a subgroup of ΓF of finite index t. We use theprevious notation, but now we have N ⊂ G′ ⊂ G. Thus, ρ∞ and ρ ′∞ are the restrictions of ρ and ρ ′ toN, respectively. In this case, ρ∞ is a direct sum of representations obtained from ρ ′∞ by composing withcertain automorphisms of N. The automorphism are just the restriction of certain inner automorphismsof G, namely the inner automorphisms defined by some set of coset representatives g1, ...,gt for G′ in G.We take g1 to be the identity. Denote these representations of G′ by ρ ′1, ...,ρ

′t . (They are not necessarily

distinct.) Define the corresponding discrete G′-modules D′1, ...,D′t obtained from D′ by composing with the

above specified automorphisms of N. We have D′1 = D′. For each i, 1≤ i≤ t, conjugating by gi also definesan isomorphism of H1(N,D′) to H1(N,D′i). Since ΓF is commutative, this isomorphism is ΓF ′ -equivariant.Also, the isomorphism induces an isomorphism of SelD′(F∞) to SelD′i(F∞). Thus, the SelD′i(F∞)’s are allisomorphic to SelD′(F∞) as Λ(O,F ′)-modules.

As a GF∞-module, D∼=⊕1≤i≤tD′i. Furthermore, D′i = gi(D′). Thus,

SelD(F∞)∼= ⊕1≤i≤t

SelD′i(F∞)

as ΓF ′ -modules and the action of ΓF permutes the summands in the corresponding way. The same thing istrue for the Pontryagin duals. It follows that

XD(F∞) ∼= XD′(F∞)⊗Λ(O,F)′Λ(O,F)

and therefore Iρ = Iρ ′Λ(O,F) as stated. ut


Propositions 3 and 4 show that it is enough to consider Iτ and Jτ where τ is a totally even, absolutelyirreducible Artin representation of GQ. For if ρ is a totally even Artin representation over F , then IndQ

F (ρ)is a direct sum of absolutely irreducible Artin representations which must also be totally even. The field E

must be chosen to be sufficiently large. Both ρ and all of the absolutely irreducible constituents of IndQF (ρ)

must be realizable over E. The next remark shows that the choice of E is otherwise not too significant.

Remark 5. Suppose that E′ and E are finite extensions of Qp with rings of integers O′ and O, respectively.Assume that E′ ⊆ E. Let Γ = Zp. Let Λ ′ =O′[[Γ ]] and Λ =O[[Γ ]], and let L′ and L be their fraction fields.Thus, Λ is the integral closure of Λ ′ in L. Since Λ ′ is integrally closed in L′, one has Λ ′ =Λ ∩L. It followsthat if I′ is a principal ideal in Λ ′, and I = I′Λ , then one can recover I′ from I by I′ = I∩Λ ′.

In particular, suppose that ρ ′ is a totally even Artin representation of GF over E′, and D′ is the corre-sponding discrete Galois module. Extending scalars to E, one obtains an Artin representation ρ over E, andone can take D = D′⊗O′ O as the corresponding discrete Galois module. One sees easily that SelD(F∞) ∼=SelD′(F∞)⊗O′ O. The characteristic ideals are related by Iρ = Iρ ′Λ(O,F). Hence Iρ ′ = Iρ ∩Λ(O′,F). Similarstatements hold for the ideals Jρ ′ and Jρ .

Our final result in this section concerns the effect of twisting the Galois representation ρ . Suppose thatξ is a 1-dimensional Artin representation of GF which factors through ΓF . We must choose E sufficientlylarge so that ξ has values in E. Thus, ξ : ΓF → O× is a continuous homomorphism. We denote the twistρ⊗ξ simply by by ρξ . The corresponding discrete GF -module is D⊗ξ = D⊗OO(ξ ), where O(ξ ) is thefree O-module of rank 1 on which GF acts by ξ . We use a similar notation below for other discrete andcompact O-modules. For brevity, we denote D⊗ ξ by Dξ . Of course, as O-modules and GF∞

-modules, wecan identify Dξ and D. The actions of ΓF on the corresponding Galois cohomology groups are related by

H1(FΣ/F∞,Dξ ) ∼= H1(FΣ/F∞,D)⊗ξ

and therefore we have the following ΓF -equivariant isomorphism of discrete O-modules:

SelDξ(F∞) ∼= SelD(F∞)⊗ξ . (6)

Both of these O-modules are Λ(O,F)-modules and the isomorphism is a Λ(O,F)-module isomorphism.

It follows from (6) that XDξ(F∞)∼= XD(F∞)⊗ξ−1 as Λ(O,F)-modules. Furthermore, noting that

H0(FΣ/F∞,Dξ ) = H0(FΣ/F∞,D)⊗ξ ,

it follows that YDξ(F∞)∼=YD(F∞)⊗ξ−1 as Λ(O,F)-modules. These isomorphisms give a simple relationship

between Iρξand Iρ and between Jρξ

and Jρ , as we now discuss.

In general, suppose that Γ is a commutative pro-p group. Let ΛO =O[[Γ ]]. We have the natural inclusionmap ε : Γ → Λ×. Suppose that ξ : ΓF → O× is a continuous homomorphism. Since O× ⊂ Λ

×O , we obtain

a continuous homomorphism ξ ε : Γ → Λ×O . This is the map γ 7→ ξ (γ)γ for all γ ∈ Γ . We can then extend

ξ ε to a continuous O-algebra homomorphism twξ : ΛO→ΛO. This is an automorphism of ΛO. The inversemap is twξ−1 .

Now suppose that Γ ∼= Zp and that ξ : Γ → O× is a continuous homomorphism. Let X be a finitely-generated, torsion ΛO-module. For any λ ∈ ΛO, X [λ ] denotes x ∈ X | λx = 0 . The characteristic idealIX is determined by the invariants µ(X) and rankO

(X [θ t ]

), where θ varies over the irreducible elements in

ΛO and t ≥ 1. We write Xξ−1 for X ⊗ξ−1. It is easily seen that µ(X) = µ(Xξ−1). We regard X and Xξ−1 asthe same O-modules, but with different O-linear actions of Γ . If γ ∈ Γ , and x ∈ X = Xξ−1 , we denote thefirst action by γ ·x and the second by γ · ·x. Thus, γ · ·x = ξ (γ)−1γ ·x. That is, we have (ξ (γ)γ) · ·x = γ ·x forall γ ∈ Γ ,x ∈ X . It follows that twξ (θ) · ·x = θ ·x for all θ ∈ΛO and x ∈ X . In particular, for any irreducibleelement θ ∈ΛO and t ≥ 1, we have

Xξ−1 [twξ (θt)] = X [θ t ] .

Consequently, we have the following result.

172 Ralph Greenberg

Proposition 5. Let ξ be a character of finite order of ΓF . Then Iρξ = twξ (Iρ) and Jρξ = twξ (Jρ).

Remark 6. In the introduction, we mentioned the Selmer group SelD(n)(F∞) associated to the Tate twistD(n) when n≥ 2 and n≡ 0 (mod d), where d = [F(µ2p) : F ]. Note that χn

F = κnF factors through ΓF for any

such n. The above discussion shows how the actions of ΓF on SelD(F∞) and SelD(n)(F∞) differ. In fact, if θ

generates the characteristic ideal of XD(F∞), then twκnF(θ) generates the characteristic ideal of XD(n)(F∞).

3 The definition of p-adic Artin L-functions.

If F is totally real and ρ : GF → GLd(C) is a totally even Artin representation, then the correspondingArtin L-function L(z,ρ) is a meromorphic function on C. We will let L∗(z,ρ) denote the function given bythe same Euler product as L(z,ρ), but with the Euler factors for the primes of F lying above p omitted. Atheorem of Siegel implies that L(1− n,ρ) ∈ Q(ψ) for all integers n ≥ 1. Here ψ is the character of ρ andQ(ψ) is the field generated by its values. Furthermore, L(1−n,ρ) 6= 0 when n is even. For our purpose, wewill consider the values L∗(1−n,ρ). They are also algebraic numbers and are nonzero for even n≥ 2.

The above L-values behave well under conjugacy in the following sense. If g ∈ Gal(Q/Q), then ψ ′ =g ψ is the character of another totally even Artin representation ρ ′ of GF . We then have L(1− n,ρ ′) =g(L(1−n,ρ)

)for all n≥ 1. The Euler factors for primes above p behave similarly, and so we have the same

conjugacy properties for the values L∗(1−n,ρ). Therefore, if we arbitrarily choose embeddings of Q intoC and into Qp, then the complex algebraic numbers L∗(1− n,ρ) and the values of ψ can all be regardedas elements of Qp. The character ψ is then the character of an Artin representation ρ : GF → GLd(Qp). Ofcourse, ψ determines ρ up to equivalence. The values L∗(1− n,ρ) are also determined by the Qp-valuedcharacter ψ , and do not depend on the choice of embeddings. In fact, if ψ has values in a finite extension E

of Qp, then the values L∗(1−n,ρ) for n≥ 1 are also in E, and are nonzero when n is even.These L-values also behave well under induction. Suppose that F ′ is a finite, totally real extension of

F . Suppose that ρ = IndFF ′(ρ

′), where ρ ′ is a totally even Artin representation of GF ′ . We have L(z,ρ) =L(z,ρ ′). The same identity is also true even if we delete the Euler factors for primes above p. Thus,

L∗(1−n,ρ) = L∗(1−n,ρ ′)

for all n≥ 1.The p-adic L-function Lp(s,ρ) satisfies the following interpolation property:

Lp(1−n,ρ) = L∗(1−n,ρ)

for all n ≡ 0 (mod [F(µp) : F ]) if p is odd, or all n ≡ 0 (mod 2) if p = 2. It is a meromorphic functiondefined on a certain disc D in Qp. The existence of such a function was proved by Deligne and Ribet whenρ is of dimension 1. In this case, it is holomorphic on D, except possibly at s = 1.

Suppose now that ρ factors through ∆ = Gal(K/F), where K is a finite, totally real, Galois extensionof F . The existence of Lp(s,ρ) then follows if ρ is induced from a 1-dimensional representation ρ ′ of asubgroup ∆ ′ of ∆ . Then ρ is a so-called monomial representation. If ∆ ′ = Gal(K/F ′), then ρ = IndF

F ′(ρ′)

and we have Lp(s,ρ) = Lp(s,ρ ′). Thus, Lp(s,ρ) is again holomorphic on D, except possibly at s = 1.In general, a theorem of Brauer states that there exist monomial representations ρ1, ...,ρs σ1, ...,σt of ∆ ,

where s, t ≥ 0, such that

ρ⊕( t⊕j=1

σ j)∼= s⊕

i=1ρi (7)

and so we can define Lp(s,ρ) as the quotient ∏si=1 Lp(s,ρi)

/∏

tj=1 Lp(s,σ j). The above interpolation prop-

erty is indeed satisfied by this function.

Let Γ = ΓQ = Gal(Q∞/Q). There is a canonical isomorphism κ : Γ → 1+qZp, where q = p for odd pand q = 4 for p = 2. It is defined as the composite map

Gal(Q(µp∞)/Q)χ−→Z×p −→ 1+qZp


which indeed factors through Γ . Here χ is the p-power cyclotomic character. The second map is just theprojection map for the decomposition Z×p =W × (1+qZp), where W is the group of roots of unity in Qp.For any s∈Zp, one can define κs : Γ → 1+qZp, which is a continuous group homomorphism. It extends toa continuous O-algebra homomorphism Λ(O,Q)→ O which we also denote by κs. Furthermore, for any F ,the restriction map ΓF → Γ defines an injective homomorphism Λ(O,F)→Λ(O,Q). We identify Λ(O,F) withits image and define κs

F to be the restriction of κs to that subring.Let L(O,F) denote the fraction field of Λ(O,F). Suppose that θ ∈ L(O,F). Write θ = αβ−1, where α,β ∈

Λ(O,F) and β 6= 0. The Weierstrass preparation theorem implies that κsF(β ) 6= 0 for all but finitely many

s ∈ Zp. Thus, excluding a finite set of values of s, one can make the definition κsF(θ) = κs

F(α)κsF(β )

−1.Furthermore, one has the following property:

If θ1, θ2 ∈ L(O,F) and κsF(θ1) = κs

F(θ2) for infinitely many s ∈ Zp, then θ1 = θ2.

One verifies this by writing θ1 = α1β−11 , θ2 = α2β

−12 , and applying the Weierstrass preparation theorem to

α1β2−α2β1.

One can associate to Lp(s,ρ) a nonzero element θρ of L(O,F). It is characterized as follows:

Lp(1− s,ρ) = κsF(θρ) for all but finitely many s ∈ Zp. (8)

for all but finitely many s ∈ Zp. If ρ is 1-dimensional, then Deligne and Ribet’s construction of Lp(1− s,ρ)proves the existence of such a θρ . Furthermore, they show that

Jρ θρ ⊆ 2[F :Q]Λ(O,F) ,

where Jρ is the ideal in Λ(O,F) defined in the introduction. Note that Jρ = Λ(O,F) unless ρ factors throughΓF . If ρ is monomial, one can use proposition 4 to prove that θρ exists. Then θρ has the integrality property

Jρ θρ ⊆ 2[F :Q]deg(ρ)Λ(O,F) (9)

since if ρ is induced from a 1-dimensional Artin representation ρ ′ of GF ′ , where F ′ is a finite extension ofF , then [F ′ : Q] = [F : Q]deg(ρ).

If ρ has arbitrary dimension, then the existence of θρ satisfying (8) follows from (7). One assumes atfirst that E is large enough so that all of the monomial representations ρi and σ j are realizable over E. Theθρi ’s and θσ j ’s are nonzero elements in the fraction fields of various subrings of Λ(O,F). One can then define

θρ =s

∏i=1

θρi

/ t

∏j=1

θσ j (10)

With this definition, we can only say that θρ is an element in the fraction field of Λ(O,F).The behavior of the values L∗(1− n,ρ) under conjugacy implies a similar behavior for the elements

θρ . To be precise, suppose that γ ∈ GQp . Let O′ = γ(O). Let ρ ′ = γ ρ . Note that γ induces a continuousisomorphism from Λ(O,F) to Λ(O′,F). This isomorphism extends to an isomorphism of the fraction fields,which we also denote by γ . We then have θρ ′ = γ

(θρ

).

Concerning the choice of O, the above conjugacy property and a straightforward Galois theory argumentshow that one can even take O to be the extension of Zp generated by the values of the character of ρ . Inthe next section, we will see that the integrality property (9) still holds when p is odd.

The above remarks give us the following properties of the θρ ’s which are parallel to the assertions inpropositions 3 and 4.

Proposition 6. With the same notation as in proposition 3, we have θρ = θρ1θρ2 .

Proposition 7. With the same notation as in proposition 4, we have θρ = θρ ′ .

174 Ralph Greenberg

We will also need the analogue of proposition 5. It relies on another property of the p-adic L-functionsconstructed by Deligne and Ribet. The interpolation property for θρ stated before can be expressed asfollows:

κnF(θρ) = L∗(1−n,ρ) = L∗(1,ρκ

nF)

for all n ≥ 2 satisfying n ≡ 0 (mod p− 1) if p is odd (or n ≡ 0 (mod 2) if p = 2). The underlying E

representation space for ρκnF is the Tate twist V (n). However, if ξ is a character of ΓF of finite order, and O

contains the values of ξ , then Deligne and Ribet also show that

κnF ξ (θρ) = L∗(1−n,ρξ ) = L∗(1,ρξ κ

nF) = L∗(1,ρκ

nF ξ ).

Furthermore, one has κnF(θρξ ) = L∗(1,ρξ κn

F). Thus, we have κnF ξ (θρ) = κn

F(θρξ ) for the above values ofn.

Suppose that ϕ : ΓF → O× is any continuous homomorphism. Let ξ be as above. Then both ϕ andϕξ : ΓF → O× extend to continuous O-algebra homomorphisms ϕ and ϕξ from Λ(O,F) to O. We alsohave the continuous O-algebra homomorphism ϕ twξ : Λ(O,F)→ O. Such O-algebra homomorphisms aredetermined uniquely by their restrictions to ΓF . Note that

(ϕ twξ )(γ) = ϕ(ξ (γ)γ) = (ϕξ )(γ).

Therefore, we also have (ϕ twξ )(θ) = (ϕξ )(θ) for all θ ∈Λ(O,F). Applying this to ϕ = κnF , where n≥ 2

and n≡ 0 (mod p−1) (or n≡ 2 (mod 2) if p = 2), we obtain

κnF(twξ (θρ)

)= (κn

F twξ )(θρ) = κnF(θρξ )

for all such n and therefore it follows that twξ (θρ) = θρξ .

Proposition 8. If ξ is a 1-dimension Artin representations of GF which factors through ΓF , then θρξ =twξ (θρ).

4 Relationship of Selmer groups to p-adic L-functions

We can state the relationship quite succinctly in terms of the notation of the preceding sections. We referto this statement as the Iwasawa Main Conjecture for ρ . As before, we assume that ρ is realizable over afinite extension E of Qp with ring of integers O. Let m(ρ) = [F : Q]dim(ρ), which is just the degree of therepresentation IndQ

F (ρ).

IMC(ρ). Suppose that F is a totally real number field and that ρ is a totally even Artin representation ofGF . Then Iρ = Jρ θρ 2−m(ρ).

Note that Iρ is an ideal in Λ(O,F) by definition, but the assertion that Jρ θρ 2−m(ρ) is an ideal in that ring, andnot just a “fractional ideal”, is not at all clear from the definitions.

It is interesting to note that when p= 2, one can omit the extra power of 2 appearing in the formulation ofIMC(ρ) by merely omitting the local conditions at the archimedean primes in the definition of the Selmergroup. That is, one can define a larger Selmer group

Sel#D(F∞) = ker(

H1(F∞,D)−→ ∏η -p,∞

H1(F∞,η ,D))

.

It turns out that the characteristic ideal of the Pontryagin dual of Sel#D(F∞)/

SelD(F∞) is precisely 2m(ρ)Λ(O,F).One sees this by using the surjectivity of the map (3) together with the structure of H1

v(F∞,D) forarchimedean v as described in section 2. If I#

ρ denotes the characteristic ideal of the Pontryagin dual ofSel#D(F∞), then IMC(ρ) is equivalent to the assertion that I#

ρ = Jρ θρ .


We also consider a weaker form of IMC(ρ). It amounts to the above equality up to multiplication by apower of a uniformizing parameter π of O. We will denote the ring Λ(O,F)[

1π] by Λ ∗(O,F). It is a subring of

the fraction field L(O,F) of Λ(O,F).

IMC(ρ)∗. We have IρΛ ∗(O,F) = Jρ θρΛ ∗(O,F).

The main purpose of this section is to point out that the results proved by Wiles in [16] are sufficient toprove IMC(ρ) for all p≥ 3. Let

A(ρ) = I−1ρ Jρ θρ 2−m(ρ)

which is a principal fractional ideal of Λ(O,F), i.e., a nonzero free Λ(O,F)-submodule of the fraction fieldL(O,F). Such fractional ideals form a group.

If I is a nonzero principal ideal of Λ(O,F), and θ is a generator of I, then we will refer to µ(Λ(O,F)/I)as the µ-invariant associated to I, or to θ . We denote it by µI . For a principal fractional ideal I = I1I−1

2 , wedefine the associated µ-invariant by µI = µI1

− µI2. A simple direct argument, or the fact that Λ(O,F) is a

UFD, shows that µI is well-defined.Suppose that ρ = ρ1⊕ρ2, where ρ1 and ρ2 are totally even Artin representations of GF . It is clear that

m(ρ) = m(ρ1)+m(ρ2). It therefore follows from propositions 3 and 6 that

A(ρ) = A(ρ1)A(ρ2) . (11)

Suppose that F ′ is a finite, totally real extension of F , that ρ ′ is a totally even Artin representation ofGF ′ , and that ρ = IndF

F ′(ρ′). It is clear that m(ρ) = m(ρ ′), and so propositions 4 and 7 imply that

A(ρ) = A(ρ ′) . (12)

The conjecture IMC(ρ) asserts that A(ρ) =Λ(O,F). The conjecture IMC(ρ)∗ asserts that A(ρ) is gener-ated by a power of π . Suppose that ρ factors through ∆ = Gal(K/F), where K is totally real. It is clear from(11) and (12) that if one proves IMC(ρ ′) (respectively, IMC(ρ ′)∗) for all 1-dimensional representation ρ ′

of all subgroups ∆ ′ of ∆ , then IMC(ρ) (respectively, IMC(ρ)∗) follows.Now suppose that ξ is a 1-dimensional Artin representation which factors through ΓF . Obviously,

m(ρξ ) = m(ρ). It therefore follows from propositions 5 and 8 that

A(ρξ ) = twξ

(A(ρ)

)(13)

When p is an odd prime, Wiles proves IMC(ρ)∗ for a certain class of totally even, 1-dimensional Artinrepresentations ρ over any totally real number field F , namely the representations which factor throughGal(K/F) where K ∩ F∞ = F . (These are the representations of type S.) This result is Theorem 1.3 in[16]. Therefore, A(ρ) is generated by a power of π for all those ρ’s. However, one sees easily that ifρ is 1-dimensional, but not of type S, then there exists a 1-dimensional Artin representation ξ factoringthrough ΓF such that ρξ is of type S. Since twξ (π) = π , it follows that if A(ρξ ) = (π t), then A(ρ) = (π t).Therefore, IMC(ρ)∗ is established when p is an odd prime, F is any totally real number field, and ρ isany 1-dimensional, even Artin representation. It then follows from (11) and (12) that IMC(ρ)∗ holds for anarbitrary totally even Artin representation of GF when p is odd. For p = 2, Wiles does prove partial results,but not enough to prove IMC(ρ)∗

Now consider IMC(ρ). Wiles proves this assertion when p is odd and ρ is 1-dimensional and has orderprime to p. It follows from Theorem 1.3 and 1.4 in [16]. The above remarks and the lemma below allowone to establish IMC(ρ) for all ρ .

An alternative way to deal with the µ-invariants when ρ is 1-dimensional, but of order divisible by p,is described in [4], pages 9 and 10. It is in terms of Galois groups instead of Selmer groups. We shouldalso add that theorem 1.4 in [16] involves odd 1-dimensional representations instead of even. However,the equivalence of that theorem with what we need here is a consequence of the so-called “ReflectionPrinciple”. It is also a consequence of theorem 2 in [6].

176 Ralph Greenberg

Lemma 1. Suppose that G is a finite group and that ψ is the character of a representation of G over Qp.Assume that the values of ψ are in Qunr

p . Then, there exists subgroups Hi of G and a 1-dimensional characterψi of each Hi, where 1≤ i≤ t for some t ≥ 1, such that

1. mψ = ∑ti=1 miIndG

Hi(ψi), where m,m1, ...,mt ∈ Z, m≥ 1

2. Each ψi has order prime to p

Proof. Brauer’s theorem asserts that ψ = ∑si=1 aiIndG

Si(ϕi), where the ai’s are integers, the Si’s are sub-

groups of G, and, for each i, ϕi is a 1-dimensional character of Si. If σ ∈GQunrp , then σ fixes the values of ψ

and one also has ψ = ∑ti=1 niIndG

Si(σ ϕi). The extension of Qunr

p generated by the values of all the ϕi’s isfinite. If m is its degree, then, by taking the trace, one obtains

mψ =s

∑i=1

biIndGSi(τi)

where τi is the sum of the conjugates of ϕi over Qunrp and the bi’s are integers. Since induction is transitive,

it is enough to prove the lemma for each of the characters τi of Si. The values of τi are in Qunrp .

Let S be a subgroup of G and let ϕ be a 1-dimensional character of S. Then ϕ = αβ , where α has orderprime to p and β has p-power order. Let pk be the order of β . We can assume that k ≥ 1. The values of α

are in Qunrp . Let τϕ and τβ denote the sums of the conjugates of ϕ and of β over Qunr

p , respectively. Notethat τϕ = ατβ .

Let T = ker(β ). Thus, β factors through S/T , which is cyclic of order pk. The conjugates of β overQunr

p are the characters of S/T of order pk. Let T1 be the subgroup of S such that T ⊂ T1 ⊆ S and T1/T iscyclic of order p. Let εT and εT1

denote the trivial characters of T and T1, respectively. Then IndST (εT ) is the

sum of all the irreducible characters of S/T and IndST1(εT1

) is the sum of the irreducible characters factoring

through S/T1. Hence the difference is τβ . Thus, τβ = IndST (εT )− IndS

T1(εT1

). It follows that

τϕ = IndST (αT )− IndS

T1(αT1

),

where αT and αT1denote the restrictions of α to T and to T1, respectively. Their orders are prime to p. ut

Assume now that p is odd and that ρ is an arbitrary totally real Artin representation over F . We assumethat ρ is realizable over a finite extension E of Qp. Assume further that E is Galois over Qp. Since IMC(ρ)∗

is established, it suffices to just consider the µ-invariants to prove IMC(ρ). Now if ρ and ρ ′ are conjugateover Qp under the action of Gal(E/Qp), then one sees easily that Iρ and Iρ ′ are conjugate under the naturalaction of Gal(E/Qp) on Λ(O,F). In particular, the µ-invariants associated to those ideals are equal. The µ-invariants associated to Jρ and Jρ ′ are 0. In addition, θρ and θρ ′ are conjugate too, and so the µ-invariantsassociated to those elements of L(O,F) are equal. It follows that the µ-invariants associated to A(ρ) andA(ρ ′) are equal.

Let ρ = ⊕σ σ , where σ runs over the conjugates of ρ over Qp. Then the character of ρ has values inQp, and hence in Qunr

p . Furthermore, the above remarks show that it suffices to prove that the µ-invariantfor A(ρ) vanishes. The µ-invariant for A(ρ) will then also vanish. Lemma 1, the behavior of A(·) underinduction and direct sums, and Theorem 1.4 in [16] imply that the µ-invariant for A(ρ) is indeed zero.

Thus, we have proved the following result based on [16].

Proposition 9. Suppose that p is an odd prime. If F is any totally real number field and ρ is any totallyeven Artin representation of GF defined over a finite extension of Qp, then IMC(ρ) is true.

References

1. P. Deligne, K. Ribet, Values of abelian L-functions at negative integers over totally real fields, Invent. Math. 59 (1980),227-286.


2. D.Barsky, Fonctions zêta p-adiques d’une classe de rayon des corps totalement réels, Groupe d’étude d’analyse ultra-métrique 1977-78; errata 1978-79.

3. P. Cassou-Noguès, Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta p-adiques, Invent. Math. 51 (1979),29-59.

4. J. Coates, D. Kim, Introduction to the work of M. Kakde on the non-commutative main conjectures for totally real fields,in Noncommutative Iwasawa Main Conjectures over Totally Real Fields, Editors J. Coates, P. Schneider, R. Sujatha, O.Venjakob, Springer, 2011, 1-22.

5. R. Greenberg, On p-adic Artin L-functions, Nagoya Math. Jour. 89 (1983), 77-87.6. R. Greenberg, Iwasawa theory for p-adic representations, Adv. Stud. Pure Math. 17 (1989), 97-137.7. R. Greenberg, Iwasawa Theory, Projective Modules, and Modular Representations, Memoirs of the AMS 992 (2011),

1-185.8. R. Greenberg, V. Vatsal, On the Iwasawa invariants of elliptic curves, Invent. Math. 142 (2000), 17-63.9. R. Greenberg, V. Vatsal, Iwasawa Theory for Artin representations, in preparation.

10. B. Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183-266.11. B. Mazur, A. Wiles, Class fields for abelian extensions of Q, Invent. Math. 76 (1984), 179-330.12. J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, Grundlehren der Math. Wissenschaften 323 (2000),

Springer.13. B. Perrin-Riou, Variation de la fonction L p-adique par isogénie, Adv. Stud. Pure Math. 17 (1989), 347-358.14. K. Ribet, Report on p-adic L-functions over totally real fields, Astérisque 61 (1979), 177-192.15. P. Schneider , The µ-invariant of isogenies, Jour. of the Indian Math. Soc. 52 (1987), 159-170.16. A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. 131 (1990), 493-540.

Iwasawa µ-invariants of p-adic Hecke L-functions.

Ming-Lun Hsieh

Abstract This article surveys recent developments on Iwasawa µ-invariants of p-adic Hecke L-functionsfor CM fields following Hida.

1 Introduction

1.1 p-adic Hecke L-function for CM fields

This aim of this paper is to report recent results on the vanishing of µ-invariants of p-adic Hecke L-functionsfor CM fields built upon fundamental works of Hida in [Hid10]. The significance of these results stems fromtheir applications to Iwasawa main conjectures for CM fields and CM elliptic curves over totally real fields.We begin with some notation to introduce p-adic L-functions. Let p > 2 be an odd rational prime. Let Fbe a totally real field of degree d over Q and K be a totally imaginary quadratic extension of F . Let DFbe the absolute discriminant of F . Fix two embeddings ι∞ : Q→ C and ιp : Q→ Cp once and for all. Letc denote the complex conjugation on C which induces the unique non-trivial element of Gal(K/F). Let Kbe the composite of all Zp-extensions of K. Then K is Galois over F , and the complex conjugation acts onGal(K/K) by the usual conjugation. Put

G+ :=

σ ∈ Gal(K/K) | cσc = σ

.

Denote by K−∞ the fixed subfield of G+ in K. Then the Galois group Gal(K−∞ /K) is a free Zp-module ofrank d by class field theory. We call K−∞ the anticyclotomic Zd

p-extension of K. On the other hand, let Q∞

be the cyclotomic Zp-extension of Q, and denote by K+∞ := KQ∞ the cyclotomic Zp-extension of K. Let

K∞ = K+∞ K−∞ be the composite of K+

∞ and K−∞ . If Leopoldt conjecture holds for F , then K∞ = K. Let W bethe ring of integers of a finite extension of the p-adic completion of the maximal unramified extension ofQp. Define

Λ• :=W [[Gal(K•∞/K)]] (•= /0,+,−).

Then Λ = W [[Gal(K∞/K]] is an Iwasawa algebra over W of (d + 1)-variables. Fix a set of the generatorsγ(i)−

i=1,··· ,d

(resp. γ+) of Gal(K−∞ /K) (resp. Gal(K+∞ /K)). The restriction maps Gal(K∞/K)→Gal(K±∞ /K)

induce the isomorphismΛ = Λ+⊗W Λ− =W [[T+,S1, · · · ,Sd ]]

with the cyclotomic variable T+ = γ+−1 and the anticyclotomic variables

Si = γ(i)− −1

i=1,··· ,d

.

We assume the following ordinary hypothesis:

Ming-Lun HsiehDepartment of Mathematics, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan, e-mail:[email protected]

179

[email protected]

180 Ming-Lun Hsieh

Every prime of F above p splits in K. (ord)

Let Σ be a p-adic CM type of K. In other words, Σ is a set of p-adic places of K such that Σ and its complexconjugation Σc are disjoint and Σ

⊔Σc is the set of all places of K above p. The existence of a p-adic CM

type Σ is assured by the ordinary assumption (ord). Let Σ∞ be the CM type corresponding to Σ, i.e. Σ is theset of embeddings σ : K → Q such that ιp σ induces a p-adic palace in Σ. Let χ be a Hecke character of Kof infinity type kΣ∞ with k ≥ 1. Suppose that χ takes value in W . According to [Kat78] and [HT93], thereexists a unique element Lχ,Σ ∈W [[Gal(K∞/K)]] characterized by the following interpolation property: forevery finite order characters ν : Gal(K∞/K)→ µp∞ of conductor ∏P|pP

nP ,

ν(Lχ,Σ) =Γ (k)dL(0,χν)

Ω k · ∏P∈Σp

eP(χν) ·[O×K : O×F ]√

DF. (1)

where (i) eP(χν) is the p-adic multiplier defined by

eP(χν) =

(1−χν(P))(1−χ−1ν−1(P)) if nP = 0,τ(χPνP) if nP > 0.

Here νP is the character of ν restricted on the completion K×P via Artin reciprocity law, and τ(χPνP) isthe Gauss sum of χPνP ([Lan94, p.286]) (ii) Ω ∈ C× is the complex CM period of K ([Kat78, (5.1.45)]).We call Lχ,Σ the (primitive) p-adic Hecke L-function attached to the character χ . Define the cyclotomic andanticyclotomic p-adic L-functions attached to χ by

L+λ ,Σ := Lχ,Σ|S1=...=Sd=0 ∈Λ+, L−

χ,Σ := Lχ,Σ|T+=0 ∈Λ−.

1.2 The µ-invariants

We are interested in the µ-invariants of these p-adic L-functions. Recall that the µ-invariant µ(L•χ,Σ) is

defined byµ(L•χ,Σ) = max

r ∈Q≥0 | L•χ,Σ ∈ pr

Λ•.

Hida in [Hid10] invented a method to compute the µ-invariants of Lχ,Σ and L−χ,Σ at least when p is unramified

in F . For example, it is proved in [BH13] and [Hid11] that

Theorem 1. If p - DF , then µ(Lχ,Σ) = 0.

In other words, Lχ,Σ 6≡ 0(mod mW Λ), where mW is the maximal ideal of W . However, the determination ofthe µ-invariant of the cyclotomic p-adic L-function L+

χ,Σ unfortunately seems out of reach with the currenttechniques.

We describe the result for anticyclotomic p-adic L-functions after introducing some notation. Let C bethe prime-to-p conductor of χ and let C− be the product of non-split prime divisors of C. For each primedivisor q|C−, we define the local invariant µp(χq)≥ 0 by

µp(χq) := infx∈K×q

ordp(χq(x)−1),

where ordp : C×p →Q is the p-adic valuation normalized so that ordp(p) = 1. Put

µp(χ) := ∑q|C−

µp(χq).

We say χ is self-dual if χ|A×F = τK/F |·|AF, where τK/F is the quadratic character associated to K/F . It is

known that the complex L-function L(s,χ) of a self-dual character satisfy the function equation L(s,χ) =NsW (χ)L(−s,χ), where N is a positive integer and W (χ) ∈ ±1 is called the global root number of χ . Itfollows from the functional equation that L−

χ,Σ = 0 if W (χ) =−1. On the other hand, we say χ is residually

Iwasawa µ-invariants of p-adic Hecke L-functions. 181

self-dual if χ(a)NF/Q(a)≡ τK/F(a)(mod mW ) for all prime-to-pC ideals a of OF . The following theoremsare proved in [Hid10] if C− = (1) and [Hsi13] for the general case.

Theorem 2. Suppose that p - DF . Let χ be a self-dual Hecke character such that the global root numberW (χ) = 1. Then we have

µ(L−χ,Σ) = µp(χ).

Theorem 3. Suppose that p - DF and that χ is not residually self-dual. Then

µ(L−χ,Σ) = 0 ⇐⇒ µp(χ) = 0.

Remark 1. When F = Q, Theorem 2 was also proved by Finis [Fin06] by a different approach.

If χ is self-dual with the root number W (χ) =−1, then Lχ,Σ has at least a simple zero at the cyclotomicvariable T+ = 0 by the functional equation of L-functions for CM fields. Thus the anticyclotomic p-adic L-function L−

χ,Σ vanishes. In this case, we can consider the cyclotomic derivative

L(1),−χ,Σ :=

∂Lχ,Σ

∂T+|T+=0 ∈Λ−.

Let h−K be the relative class number of K/F . In [Bur14], A. Burungale uses Hida’s method and some inputsfrom [Hsi13] to prove the following result.

Theorem 4. If p - h−K ·DF and χ is self-dual with W (χ) =−1, then

µ(L(1),−χ,Σ ) = µp(χ)+ min

q|C−

0,µ ′p(χq)

,

where

µ′p(χq) := ordp(

logp NK/Q(q)

logp(1+ p))−µp(χq)

and logp : Z×p → Zp is the p-adic logarithm, which is zero on the roots of unity and defined by the usualpower series on 1+ pZp.

In particular, if µp(χ) = 0, then µ(L(1),−χ,Σ ) = 0.

1.3 p-adic L-functions and Selmer groups for CM elliptic curves

The p-adic L-functions for CM fields allow us to construct p-adic L-functions for CM elliptic curves overF . Let E be an elliptic curve defined over F . Suppose that E admits complex multiplication by the ringof integers of an imaginary quadratic field Q(

√−D) with the absolute discriminant D > 0 and that E has

good ordinary reduction at all places above p. In particular, this implies that p = pp is split in Q(√−D).

Let K be the CM field F(√−D). Let Σp be the set of primes of K above p. Then Σp is a p-adic CM type of

K. For each P ∈ Σp, let kP be the residue field and N(P) = #(kP). Let αP be the p-adic unit root of theHecke polynomial X2−aP(E)X +N(P), where aP(E) = 1+N(P)−#(E(kP)). By the theory of complexmultiplication, we can associate a Hecke character χE of K to the CM elliptic curve E/F so that χ

−1E is

self-dual, and the Hecke L-function L(s,χE) of χE is equal to the Hasse-Weil L-function L(s,E/F) of E/F .We define

LE,p := Lχ−1E ,Σp

∈Λ

to be the p-adic Hecke L-function attached to χ−1E . The p-adic L-function for E/K∞ is defined by

Lp(E/K∞) := (LE,p)2.

It follows from (1) that for every finite order character ν : Gal(K∞/K)→ µp∞ , we have

182 Ming-Lun Hsieh

ν(Lp(E/K∞)) =L(1,E/K⊗ν)

Ω 2E

· ∏P∈Σp

eP(E,ν)2 ·[O×K : O×F ]

2

DF,

where ΩE = ∏σ∈Σ ΩEσ is a product of the periods ΩEσ of a Néron differential form of Eσ over OFσ ,(p) andeP(E,ν) = eP(χ−1

E ν).Iwasawa main conjecture for E/K∞ asserts that the p-adic L-function Lp(E/K∞) is given by the charac-

teristic power series of the p-Selmer group of E/K∞. We recall that for each algebraic extension L of F , thep-Selmer group Selp(E/L) of E over L is defined by

Selp(E/L) = ker

H1(L,E[p∞])→∏v

H1(Lv,E),

where v runs over all places of L. We denote by Xp(E/L) the Pontryagin dual of Selp(E/L). If L is a Zrp-

extension of K, then the Iwasawa algebra Zp[[Gal(L/K)]] is a formal power series ring of r variables, andXp(E/L) is a finitely generated Zp[[Gal(L/K)]]-module. Let

Cp(E/L) ∈ Zp[[Gal(L/K)]]

be the characteristic power series of the module Xp(E/L) over Zp[[Gal(L/K)]]. Recall that the character-istic power series of a finitely generated Zp[[Gal(L/K)]]-module M is a generator of the intersection of allprincipal ideals containing the Fitting ideal of M. Now we can state Iwasawa main conjecture for E/K∞ asfollows:

Conjecture 1. Suppose that p is unramified in F . Then the principal ideal Cp(E/K∞)Λ is generated byL(E/K∞). 1

Define the cyclotomic and anticyclotomic p-adic L-functions by L+E,p = L+

χ−1E ,Σp

∈Λ+ and L−E,p = L−χ−1E ,Σp

∈

Λ−. Similarly, define L(E/K±∞ ) = (L±E,p)2, and we can formula the main conjectures for these p-adic L-

functions in the same way. When F = Q, there is a completely different construction of LE,p by Yager[Yag82], combined with which the main conjectures has been settled down by Rubin [Rub91].

We discuss the connection between the main conjectures for CM elliptic curves and the µ-invariants ofp-adic L-functions.

Theorem 5. Suppose that p - DF . We have

(1) µ(LE,p) = 0,(2) if the root number W (E/F) = +1, then µ(L−E,p) = 0.

Proof: Part (1) is an immediate consequence of Theorem 1. To see part (2), we note that W (E/F) =W (χ−1E )

and χE takes value in the imaginary quadratic field Q(√−D). Therefore, χE,q|OK×v

takes value in the group

of the roots of unity in Q(√−D) for every finite place q of K. In particular, this implies µp(χE,q) = 0 for

every q|C− whenever p - 2D, so µp(χE) = 0, and (2) follows from Theorem 2. ut

Remark 2. When F = Q, the µ-invariants of LE,p along the Zp-extension unramified outside p were provedby Gillard and Schneps ([Gil87] and [Sch87]). In particular, these results also imply µ(LE,p) = 0.

If the root number W (E/F) = −1, then LE,p has at least a simple zero at the cyclotomic variable T+ = 0,and Selp(E/K−∞ ) is expected to have corank two over Λ−. In this case, we consider the derivative

L(1),−E,p :=

∂LE,p

∂T+|T+=0 ∈Λ−.

We have seen µp(χE) = 0, so the following is an immediate consequence of Theorem 4.

Theorem 6. If p - DF ·h−K and W (E/F) =−1, then µ(L(1),−E,p ) = 0

1 Note that the p-adic L-function Lp(E/K∞) only belongs to Λ instead of Zp[[Gal(K∞/K)]]. In addition, p is assumed to beunramified in F , since it is not clear to the author that the Katz p-adic L-function Lp(E/K∞) is the right one for the mainconjecture if p is ramified in F .


Theorem 6 implies that L(1),−E,p is non-zero and hence LE,p does have a simple zero at T+ = 0. When F = Q,

the non-vanishing of L(1),−E,p was proved in [AH06] by an algebraic method.

The vanishing of µ-invariants of the anticyclotomic p-adic L-functions L−E,p and L(1),−E,p plays a crucial

role in the Eisenstein ideal approach to a one-sided divisibility towards Iwasawa main conjecture for CMfields. It has the following application to the cyclotomic main conjecture for CM elliptic curves over totallyreal fields [Hsi14, Thm. 1].

Theorem 7. Suppose that p - 3DF ·h−K . Then

Lp(E/K+∞ ) divides Cp(E/K+

∞ ) in Λ+.

We explain very briefly how the vanishing of µ-invariants enters into the proof of the above theorem. In[Hsi14], we study the main conjecture by the Eisenstein congruence method for the quasi-split unitary groupU(2,1) of degree three over F . The general philosophy is that some p-adic L-functions can be realized as theconstant term of a particular p-adic family of Eisenstein series, and the congruence between cusp forms andthis special Eisenstein series modulo its constat term yields elements in the Selmer groups correspondingto p-adic L-functions. In our setting, we construct a Λ -adic Eisenstein series on U(2,1)/F whose constantterm is a product of LE,p and a Deligne-Ribet p-adic L-function LDR, which should contribute to two distinctSelmer groups respectively. However, it turns out that our method does not allow us to distinguish theircontributions to individual Selmer group unless we know the common zeros of LE,p and LDR are at mostsimple zeros. Now if W (E/F) = +1, µ(L−E,p) = 0 implies LE,p and LDR have no common zeros, while if

W (E/F) = −1, µ(L(1),−E,p ) = 0 implies that LE,p and LDR have at most one simple zero! Therefore, we can

complete the argument to show one-sided divisibility Lp(E/K∞)|Cp(E/K∞), from which Theorem 7 followsby Iwasawa descent for CM elliptic curves due to Perrin-Riou.

In the remainder of this article, we will first review Sinnott’s proof of Ferrero-Washington theorem, andin §3 we explain Hida’ method of proving the part 2 of Theorem 5 with emphasis on the autmorphic side ofthe proof. In the last section §4, we discuss the proof of Theorem 6 and the generalization of Hida’s ideasto a class of p-adic Rankin-Selberg L-functions.

Acknowledgement. The author would like to thank the referee for many suggestions on the improvementof the manuscript.

2 Ferrero-Washington theorem

While the focus of the research reported here is about the µ-invariants of p-adic Hecke L-functions forCM fields, it is instructive to first review the case of p-adic L-functions for Dirichlet characters. Let χ :(Z/NZ)×→ C× be an odd Dirichlet character of conductor N and let c > 1 be an integer prime to p. LetO be a finite extension of Zp containing values of χ and let mO be the maximal ideal of O. Let Lχ bethe Kubota-Leopoldt p-adic L-function associated to χ and c. Namely, Lχ is a O-valued p-adic measure on1+ pZp such that for all odd positive integers k,∫

1+pZp

xkdLχ(x) = (1−χω−k(c)ck+1)L(−k,χω

−k). (1)

Recall that the µ-invariant µ(ϕ) of a Zp-valued p-adic measure ϕ on a p-adic compact group H is definedto be

µ(ϕ) = infU⊂H open

ordp(ϕ(U)).

The vanishing of the µ-invariant of Lχ was proved by Ferrero and Washington.

Theorem 8 ([FW79]). µ(Lχ) = 0.

In [Sin84], Sinnott gave a beautiful new proof of Ferrero-Washington theorem, which is the origin of Hida’smethod. Consider the torus Gm/O = SpecO[t, t−1] over O and its formal completion Gm/O = SpfO[[t−1]]

184 Ming-Lun Hsieh

at the maximal ideal m := (p, t− 1). Each a ∈ Z×p induces an automorphism on Gm/O by sending t 7→ ta.Let F = Fp. The first ingredient in Sinnott’s proof is the following linear independence result on rationalfunctions on Gm/F.

Proposition 1 (Prop. 3.1[Sin84]). Let a1, . . . ,an ∈ Z×p such that aia−1j 6∈ Z×

(p). If f1, . . . , fn ∈ F(t) are non-constant functions, then f1(ta1), . . . , fn(tan) ∈ F((t−1)) are linearly independent over F.

We can associate a formal function Gχ(t) ∈ O[[t−1]] on Gm/O to Lχ defined by

Gχ(t) :=∫

1+pZp

txdLχ(x) = ∑n≥0

(∫1+pZp

(xn

)dLχ(x)

)(t−1)n.

Thus, Gχ(t) is the usual power series attached to the measure on Zp obtained by the extension of Lχ by zerooutside 1+ pZp. Then it is well known that

µ(Lχ) = 0 ⇐⇒ Gχ(t)(mod mO) 6= 0 ∈ F[[t−1]].

The second ingredient is the expression of Gχ(t) in terms of a linear combination of a rational functions onGm acted by Z×p . To be precise, let ∆ = µp−1 be the torsion subgroup of Z×p and let ∆ alg :=∆ ∩Z×

(p) = ±1.Let D be a set of representative of ∆/∆ alg = µp−1/±1 in µp−1. Then we have

Proposition 2. Denote by O[t, t−1](m) the localization of O[t, t−1] at m.

(i) There exist a set Faa∈D of rational functions in W [t, t−1](m) indexed by D such that

Gχ(t) = ∑a∈D

χ(a−1)Fa(ta); (2)

(ii) Fa(t)(mod mO) is not a constant function for some a ∈ D.

In view of Prop. 1, this proposition implies that Gχ(t)(mod mO) 6= 0, and hence µ(Lχ) = 0.The equation (2) indeed is a consequence of the following classical formula of Dirichlet L-values:

θk(

∑Na=1 χ(a)ta

1− tN

)|t=1 = L(−k,χ), (3)

where θ := t ddt is the invariant differential operator on Gm. To illustrate this, we sketch a proof of Prop. 2 for

the special case χ = ω i with an odd integer 0 < i < p−2, where ω : (Z/pZ)×→ µp−1 is the Teichmüllercharacter. For each a ∈ Z×p , define a rational function

fa(t) =tr

1− t p , 0≤ r < p ,r ≡ a−1 (mod mO)

and putFa(t) = 2 fa(t)−2c · fca(tc) ∈ O[t, t−1](m).

Then it is straightforward to verify that

θk(∑

a∈Dχ(a−1)Fa(ta))|t=1 = (1−χω

−k(c)ck+1)L(−k,χω−k).

Hence, (2) follows from the interpolation formula (1), and the non-constancy of Fa(t)(mod mO) is clearfrom the definition.

3 Hida’s theorem on the µ-invariants of anticyclotomic p-adic Hecke L-functions

In this section, we briefly review the method developed by Hida in [Hid10] to compute the µ-invariant of theanticylotomic p-adic Hecke L-function L−

χ,Σ attached to a Hecke character χ . This method is a far-reaching


generalization of Sinnott’s proof of Ferrero-Washington theorem. We assume

p > 2 is unramified in F . (unr)

For simplicity, we further assume that µp(χ) = 0 and p does not divide the relative class number h−K ofK/F . Let O =OF and Op =OF⊗Z Zp. Fix a basis ξ1, . . . ,ξd of O over Z (d = [F : Q]). The starting pointis that, regarding the p-adic L-function L−

χ,Σ as a p-adic measure dL−χ,Σ on Gal(K−∞ /K∞)' 1+ pOp, we can

associate to L−χ,Σ a power series Gχ ∈W [[S1, . . . ,Sd ]] given by

Gχ :=∫

1+pOp

txdL−χ,Σ(x) = ∑

(k1,...,kd)∈Zd≥0

∫1+pOp

(x

k1, . . . ,kd

)dL−

χ,Σ(x)Sk11 · · ·S

kdd

(S1 = tξ1 −1, · · · ,Sd = tξd−1),

where( x

k1,...,kd

): Op→ Zp is the function defined by

(x

k1, . . . ,kd

):=

d

∏i=1

(mi

ki

)for x = m1ξ1 + · · ·+mdξd .

Then we haveµ(L−

χ,Σ) = 0 ⇐⇒ Gχ (mod mW ) 6= 0 ∈ F[[S1, · · · ,Sd ]].

To show Gχ (mod mW ) 6= 0, Hida constructs a family of p-adic Hilbert Eisenstein series Eaa∈D indexedby a suitable finite subset D of transcendental automorphisms of the deformation space of an ordinaryabelian variety A/F with CM by OK such that Gχ is a linear combination of the t-expansions of Eaa∈D atthe CM point A/F. Proving a deep result on the linear independence of p-adic modular forms modulo p (ananalogue of Prop. 1), Hida reduces the problem to showing the non-vanishing of individual p-adic Eisensteinseries Ea, which in turns is equivalent to the non-vanishing modulo p of some Fourier coefficients of Ea bythe q-expansion principle for p-adic modular forms due to Ribet. If χ is only ramified at primes split inK (this implies K/F must be unramified and W (χ) = +1 if χ is self-dual), Hida computes the Fouriercoefficients of Ea, from which he is able to easily deduce the non-vanishing of Gχ . In the general case,the computation of the Fourier coefficients of Ea is rather complicated. To resolve this difficulty, it seemstheory of automorphic representation must be brought into play. In [Hsi13], we introduce a new set ofp-adic Eisenstein series Eaa∈D, which arises from toric forms, eigenforms of Hecke operators comingfrom K×. It turns out that the connection of these Eisenstein series to p-adic L-functions is through thetoric period integral of Eisenstein series, and the non-vanishing modulo p of their Fourier coefficients areintimately related to epsilon dichotomy of the local theta correspondence for (U(1),U(1)) in [HKS96].

3.1 Linear independence for p-adic modular forms modulo p

Let N ≥ 4 be a prime-to-p integer and let c be a prime-to-p integral ideal of O. Let Lc = c∗⊕O be anO-module, where c∗ =

ξ ∈ F | TrF/Q(ξ c)⊂ Z

is the dual ideal of c. Let

M(c,N)→ SpecW

the moduli scheme of c-polarized abelian varieties with real multiplication by OF with full N-level structure.Denote by A the universal abelian scheme over M(c,N). The Igusa scheme I(c,N) is the moduli schemeover M(c,N) classifying monomorphisms ηp : c∗⊗Z µp∞/M(c,N) → A[p∞] of O-group schemes. Hence,each functorial point in I(c,N) can be written as the isomorphism class of a quintuple (A,ηp), where A =[(A,λ , ι ,ηN)] ∈M(c,N) is the datum consisting of an abelian scheme A, a c-polarization λ of A, a ringhomomorphism ι : O→ EndA and a full N-level structure ηN : Lc⊗Z Z/NZ ' A[N], and ηp : c∗⊗µp∞ →A[p∞]. Put Im = I(c,N)×SpecW SpecW/pmW for a positive integer m and let I := lim−→m

Im be the formalcompletion. Let

186 Ming-Lun Hsieh

V (c,N) = lim←−m

H0(Im,OIm)

be the space of p-adic modular forms, consisting of global sections of the formal scheme I.We introduce the t-expansion of p-adic modular forms around a CM point over F. Fix a p-ordinary CM

point x = [A] ∈M(c,N)(W ), where A/W is an abelian scheme with ordinary reduction together with thecomplex multiplication ι : OK → EndA. The reduction x0 := x⊗F lie in the ordinary locus M(c,N)ord(F).The p-adic CM-type Σ induces a decomposition OK⊗Z Zp = Op⊕Op, a 7→ (ip(a), ip(a)), so we can choosea level structure ηp,x : µp∞ ⊗ c∗ → A[p∞] at p for the CM point x such that ηp,x(ip(a)x) = ι(a)ηp,x(x) forall a ∈ OK . This gives rise to a W -point (x,ηp,x) ∈ I(c,N)(W ). Consider Sx0 the local deformation space ofx0. Namely, Sx0 is a functor on the category of local Artinian rings with residue field F and Sx0(R) is the setof the isomorphism classes [A] over R with A⊗R F ' A0. By the theory of Serre-Tate coordinate ([Kat81,Theorem 2.1]), there is a canonical isomorphism

Sx0 ' Hom(TpA0⊗Op TpA0,Gm), (1)

where TpA0 is the Tate module of the étale p-divisible group A0[p∞](F). Let T = Hom(O,Gm/W ) be thealgebraic torus over W with the character group X∗(T ) = O. Let T = Hom(Op,Gm/W ) be the associatedformal torus. Denote by t ∈ X∗(T ) the character 1 ∈ O. Then

OT = SpfW [Op] =W [[S1, · · · ,Sd ]] (Si = tξi −1).

The units group a ∈ O×p acts on OT by t 7→ ta (so Si 7→ (1+ Si)a− 1). The level structure ηp,0 at p of A0

induces an isomorphism TpA0[p∞]et 'Op. By (1), it induces an isomorphism ϕx0 : Sx0 ' T :=Hom(Op,Gm),and we obtain

ϕx0∗ : OSy' OT =W [[S1, · · · ,Sd ]]

([Hid10, Corollary 2.5]). For f ∈V (c,N), we define

f (t) := ϕx0∗( f |Sx0) ∈ OT =W [[S1, · · ·Sd ]].

The formal power series f (t) will be called the t-expansion around x0 of f .Define the subgroup Oalg

p of O×p by

Oalgp =

ip(a/a) | a ∈ O×K,(p)

.

Hida proved the following deep theorem on the linear independence of modular forms modulo p.

Theorem 9 (Corollary 3.21 [Hid10].). Let a1, · · · ,am ∈O×p such that aia−1j 6∈Oalg

p . Let f1, . . . , fm ∈V (c,N)⊗WF be non-constant functions. Then f1(ta1), . . . , fm(tam) are linearly independent over F

This theorem is an analogue of Prop. 1 in the setting of rational functions on the deformation space ofordinary CM abelan varieties. We are not able to talk about Hida’s difficult proof here, but only mentionthat two of the key ingredients: (i) a rigidity result of Ching-Li Chai on formal p-divisible groups ([Cha08,Theorem 4.3]), which implies that every closed irreducible closed subvariety Z containing z0 := (x0, . . . ,x0)in the product of ordinary locus M(c,N)ord

/F× . . .×M(c,N)ord/F is Tate linear at z0 provided that Z is stable

by the diagonal Hecke action of a finite index subgroup T of O×K,(p) (see [Hid10, Proposition 3.11] for theprecise statement); (ii) the use of Zarhin’s theorem on the Tate conjecture ([Zar75]) in the proof of [Hid10,Corollary 3.16]).

3.2 Period integrals and Fourier coefficients of toric forms

In this subsection, we explain how to combine Theorem 9 with some inputs from representation theory toshow the vanishing of µ(L−

χ,Σ) if µp(χ) = 0. Let Up be the torsion subgroup of O×p and let Ualgp =Up∩Oalg

p .


Define a subgroup T ⊂ A×K by

T :=

a ∈ A×K | a/a ∈ K×∞ O×K

.

Let Cl− := F×K×\K×/O×K be the relative class group and let Clalg− be the subgroup of Cl− generated

by the image of T in Cl−. It is easy to see that Clalg− is in fact generated by primes ramified over F . In

particular, #Clalg− is a power of 2. Since we assume p - h−K , the geometrically normalized reciprocity law

recK : K×→ Gal(Kab/K) induces an isomorphism

recp : 1+ pOp → (OK⊗Zp)×→ Gal(K−∞ /K), a 7→ recK((a,1))|K−∞ .

We define 〈·〉p : K×→ 1+ pOp by 〈a〉p = rec−1p (recK(a)|K−∞ ). Let D0 (resp. D1) be a set of representatives

of Up/Ualgp in Up (resp. Cl−/Clalg

− in K(p),×). Set

D :=D0×D1.

We rephrase the main result in [Hsi13] as follows.

Theorem 10. (i) There exists a set of p-adic modular forms Eu,a(u,a)∈D such that

Gχ(t) = ∑(u,a)∈D

χ(a)Eu,a(tu〈a〉p).

(ii) If χ is not residually self-dual or χ is self-dual with the root number W (χ) = +1, then Eu,a (mod mW )is not a constant function for some (u,a).

We have the following simple observation.

Lemma 1. The map D→ O×p /Oalgp , (u,a) 7→ u〈a〉p is injective.

Combining these with Theorem 9, we conclude that µ(L−χ,Σ) = 0 if W (χ) = +1.

We give a few words about how the proof of Theorem 10 (i) is related to the toric period integral ofEisenstein series. Fix an embedding K →M2(F) which optimally embeds OK to M2(OF). An automorphicform f : GL2(F)\GL2(AF)→ C is called a toric form of character χ if

f (gt) = χ(t) f (g) for all t ∈ T. (1)

In [Hsi13, §4], we construct some special weight one toric holomorphic Eisenstein series Ehχ,u : GL2(F)\GL2(AF)→

C for each u ∈D0. These Eisenstein series Ehχ,u are shown to be defined over W by q-expansion principle.

Let Eu,a := Ehχ,u|[a] for a ∈D1, where |[a] means the Hecke translation by a. We define the p-adic modular

form Eu,a to be the p-adic avatar of Eu,a. Put

E= ∑u∈D0

Ehχ,u ; E(t) = ∑

(u,a)∈Dχ(a)Eu,a(tu〈a〉p).

For κ ∈ Z≥0[Σ], let θ κ be the Dwork-Katz p-adic differential operator on p-adic modular forms ([Kat78,Cor. (2.6.25)]) and let δ κ

1 be the Mass-Shimura differential operator on modular forms of weight one([HT93, (1.21)]). Roughly speaking, by the interpolation property of p-adic L-functions, it is not difficult tosee that

θκ Gχ(t)|t=1 ≈ L(0,χνκ),

where νκ is some anticyclotomic Hecke character of infinity type κ(1− c), and the symbol ≈ means anequality up to periods and modified Euler factors at p and Γ -factors. The toric property (1) of these Eisen-stein series allows us to write

θκ Gχ(t)|t=1 ≈

∫A×F K×\A×K

δκ1 E(t)νκ(y)dt.

Thus, the proof of Theorem 10 boils down to the period integral formula:

188 Ming-Lun Hsieh

∫A×F K×\A×K

δκ1 E(t)νκ(t)dt ≈ L(0,χνκ),

which is proved in [Hsi13, Prop. 4.8].To show Theorem 10 (ii), we must prove the non-vanishing modulo p of some Fourier coefficient of

Eu,a(u,a)∈D. The Fourier coefficients A(ψ) of Eisenstein series are parameterized by additive charactersψ = ∏ψv : F\AF → C×, and it is a standard fact that A(ψ) = ∏v A(ψv) can be decomposed into a productof the local Whittaker integrals A(ψv), which can be determined explicitly by a straightforward computationat all places v of F unless v is non-split in K and χv is ramified, which we refer to as bad places. Take δ ∈ Kwith c(δ ) =−δ . The local Whittaker integrals A(ψv) at places those are inert or ramified in K and dividesthe conductor of χ are certain partial Gauss sum defined by

A(ψv) =∫

Fv

χv(x+δ )ψv(x)dx.

When χ is not residually self-dual, it is rather easy to show there are many A(ψ) non-vanishing modulop when µp(χ) = 0. The case χ is residually self-dual is subtle. For simplicity, we will assume that χ isself-dual with the root number W (χ) = +1. The general case can be treated with obvious modification. Tosee the role of the root number assumption, we introduce the local root number

W (χv) := ε(12,χv,ψv) ·χ∗v (−δ ) ∈ ±1 ,

where χ∗v := χv|·|− 1

2Kv

and ε(s,χ,ψv) is the ε-factor ([Tat79]). It is known that W (χv) =+1 except for finitelymany non-split places and that

W (χ) = ∏v

W (χv).

In light of epsilon dichotomy of local theta correspondence for unitary groups, the non-vanishing of theseA(ψv) is connected with the sign of W (χv). For example, using results in [HKS96, §8], we show that

W (χv) =−1⇒ A(ψv) = 0

([Hsi13, Lemma 6.1]). Therefore, we find A(ψv) = 0 for some non-split prime v if W (χ) = −1. On theother had, if W (χ) = +1, we show by a patching argument ([Hsi13, Theorem. 6.5]) that the non-vanishingmodulo p of A(ψ) for some ψ .

4 Generalizations

4.1 The cyclotomic derivative of Lχ,Σ

We consider the case where χ is self-dual with the root number W (χ) = −1 and µp(χ) = 0. In [Bur14]Burungale shows the vanishing of the µ-invariant of L(1),−

χ,Σ the anticyclotomic projection of the cyclo-tomic derivative of Lχ,Σ. The main point is that we can actually families of p-adic Eisenstein seriesEu,a(T+)(u,a)∈D over the cyclotomic variable T+ such that the t-expansion

Gχ(T+)(t) := ∑(u,a)∈D

χ(a)Eu,a(T+)(tu〈a〉p)

is the power series associated to the full p-adic L-function LE,p, and the cyclotomic derivative

∑(u,a)∈D

χ(a)E(1),−u,a (tu〈a〉p) (E

(1),−u,a :=

∂Eu,a(T+)∂T+

|T+=0)


gives rise to the power series of L(1),−χ,Σ . We can show the non-vanishing modulo p of E(1),−

u,a in a similar way

as in the case W (χ) = +1. Applying Theorem 9, we get µ(L(1),−χ,Σ ) = 0.

4.2 Anticyclotomic p-adic Rankin-Selberg L-functions

We can also apply Hida’s theorem to compute µ-invariants of a class of anticyclotomic p-adic L-functionsof Rankin-Selberg convolution of Hilbert modular forms and theta series. Let π = ⊗′πv be an irreducibleunitary cuspidal automorphic representation of GL2(AF) with central character ω and let πK the base changeto GL2(AK). Suppose that πv is a discrete series or limit of discrete series for each archimedean place v. Letχ be a Hecke character of A×K with

χ|A×F = ω−1.

Define a finite subset S(π,χ) of all places of F by

S(π,χ) :=

places v of F | ε∗(πv,χv) := ε(12,πKv ⊗χv) ·ωv(−1) =−1

.

We impose the following local root number condition:

#(S(π,χ)) = 0. (ST)

This condition holds for example when every prime divisor of the conductor of π is split in K. Then undera technical assumption on the conductor of π , in [Hsi12] we construct an anticyclotomic p-adic L-functionsLp(π,χ) ∈Λ− which interpolates the algebraic part of the Rankin-Selberg central values L( 1

2 ,πK⊗χν) foranticyclotomic Hecke characters ν of p-power conductor. The construction is similar to that of LE,p, butthis time we have to replace p-adic toric Eisenstein series by p-adic toric cusp forms

F(u,a)

(u,a)∈D. The

corresponding toric period integral is computed by an explicit version of Waldspurger formula [Wal85], andas a consequence, we obtain an expression of the power series attached to Lp(π,χ) in terms of linear com-bination of F(u,a) exactly as in Theorem 10. Using Hida’s theorem again, we can compute the µ-invariantof Lp(π,χ) by an explicit calculation of Fourier coefficients of F(u,a). We remark that the local root num-ber hypothesis (ST) is fundamental in this method since the failure of (ST) makes the toric period integralalways vanish by a theorem of Saito-Tunnell. The details can be found in [Hsi12].

Remark 3. Suppose that S(π,χ) contains all archimedean places and #(S(π,χ)) is even (so (ST) does nothold). One can still construct the anticyclotomic p-adic L-function Lp(π,χ) ∈Λ− that interpolates the alge-braic part of the central L-values L( 1

2 ,πK⊗χν) (See [BD96] and [CH12] for the elliptic case). By the workof Vatsal [Vat03], which depends on entirely different ideas from Hida’s method, and its generalization[CH13], the µ-invariant of Lp(π,χ) can be proved to vanish in many cases.

References

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BD96. M. Bertolini and H. Darmon, Heegner points on Mumford-Tate curves, Invent. Math. 126 (1996), no. 3, 413–456.BH13. A. Burungale and M.-L. Hsieh, The vanishing of µ-invariant of p-adic Hecke L-functions for CM fields, Int. Math.

Res. Not. IMRN (2013), no. 5, 1014–1027.Bur14. Ashay A. Burungale, On the µ-invariant of the cyclotomic derivative of a Katz p-adic L-function, Journal of the

Institute of Mathematics of Jussieu. (2014), DOI:10.1017/S1474748013000388.CH12. M. Chida and M.-L. Hsieh, Special values of anticyclotomic L-functions for modular forms, preprint.

ArXiv:1204.2427, 2012.CH13. M. Chida and M.-L. Hsieh, On the anticyclotomic Iwasawa main conjecture for modular forms, preprint.

ArXiv:1304.3311, 2013.Cha08. Ching-Li Chai, A rigidity result for p-divisible formal groups, Asian J. Math. 12 (2008), no. 2, 193–202.Fin06. T. Finis, The µ-invariant of anticyclotomic L-functions of imaginary quadratic fields, J. Reine Angew. Math. 596

(2006), 131–152.

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FW79. B. Ferrero and Lawrence C. Washington, The Iwasawa invariant µp vanishes for abelian number fields, Ann. ofMath. (2) 109 (1979), no. 2, 377–395.

Gil87. R. Gillard, Transformation de Mellin-Leopoldt des fonctions elliptiques, J. Number Theory 25 (1987), no. 3, 379–393.

Hid10. H. Hida, The Iwasawa µ-invariant of p-adic Hecke L-functions, Ann. of Math. (2) 172 (2010), no. 1, 41–137.Hid11. H. Hida, Vanishing of the µ-invariant of p-adic Hecke L-functions, Compos. Math. 147 (2011), no. 4, 1151–1178.HKS96. M. Harris, S. Kudla, and W. Sweet, Theta dichotomy for unitary groups, J. Amer. Math. Soc. 9 (1996), no. 4, 941–

1004.Hsi12. M.-L. Hsieh, Special values of anticyclcotomic Rankin-Selberg L-functions, preprint. ArXiv:1112.1580, 2012.Hsi13. M.-L. Hsieh, On the µ-invariant of anticyclotomic p-adic L-functions for CM fields, J. Reine Angew. Math. (2013),

DOI:10.1515/crelle-2012-0056.Hsi14. M.-L. Hsieh, Eisenstein congruence on unitary groups and Iwasawa main conjecture for CM fields, to appear in

Journal of the American Mathematical Society, 2014.HT93. H. Hida and J. Tilouine, Anti-cyclotomic Katz p-adic L-functions and congruence modules, Ann. Sci. École Norm.

Sup. (4) 26 (1993), no. 2, 189–259.Kat78. N. Katz, p-adic L-functions for CM fields, Invent. Math. 49 (1978), no. 3, 199–297.Kat81. N. Katz, Serre-Tate local moduli, Algebraic surfaces (Orsay, 1976–78), Lecture Notes in Math., vol. 868, Springer,

Berlin, 1981, pp. 138–202.Lan94. Serge Lang, Algebraic number theory, second ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New

York, 1994.Rub91. K. Rubin, The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), no. 1,

25–68.Sch87. L. Schneps, On the µ-invariant of p-adic L-functions attached to elliptic curves with complex multiplication, J.

Number Theory 25 (1987), no. 1, 20–33.Sin84. W. Sinnott, On the µ-invariant of the Γ -transform of a rational function, Invent. Math. 75 (1984), no. 2, 273–282.Tat79. J. Tate, Number theoretic background, Automorphic forms, representations and L-functions (Proc. Sympos. Pure

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Mat. 39 (1975), no. 2, 272–277, 471.

The p-adic height pairing on abelian varieties at non-ordinaryprimes

Shinichi Kobayashi

Abstract In [31] P. Schneider constructed the p-adic height pairing on abelian varieties at ordinary primesby using the universal norm subgroup. In this paper, we generalize his construction to the non-ordinarycase and compare it with that of Zarhin-Nekovár. As an application, we generalize the p-adic Gross-Zagierformula in [15] to newforms for Γ0(N) of weight 2 with arbitrary Fourier coefficients (not necessarily inQ).

1 Introduction

The theory of the p-adic valued height pairing on abelian varieties was developed in the 1980’s by Néron,Zarhin, Schneider, Mazur-Tate, etc. Compared with the real valued Néron-Tate height pairing, one importantaspect in the p-adic valued case is that the pairing depends on several choices and in this sense there is nocanonical p-adic height pairing. (One reason of this phenomenon is that there exist non-trivial compactsubgroups in Qp but not in R.) Let A be an abelian variety over a number field F . Firstly, the p-adic pairingdepends on a choice of the p-adic logarithm on the idele class group A×F /F×, which determines a Zp-extension of F . Secondly, it depends on a choice of a splitting as Qp-vector spaces of the Hodge filtration ofthe de Rham cohomology of A over the completion Fv of F at each v|p. This information is used to definethe p-adic valued local height pairing at v|p. When A has ordinary reduction at v|p, we have a natural choiceof the splitting at v obtained by the unit root subspace, and we may say there is a canonical p-adic localheight pairing at v for a fixed Zp-extension. Therefore the choice of the splitting is often made implicitly inthe ordinary case but one has to pay special attention for it in the non-ordinary case.

The canonical p-adic local height pairing at ordinary primes has a characterization by some integralproperty along the direction of the Zp-extension determined by the fixed p-adic logarithm, which is usedcrucially in the proof of the p-adic Gross-Zagier formula in [25]. This important integral property is shownby the construction of the p-adic height by using the universal norm subgroup. It is known that the universalnorm subgroup does not have sufficient information at non-ordinary primes and this makes it impossibleto construct the p-adic height via the universal norm subgroup and prevents us from investigating integralproperties of the p-adic height at non-ordinary primes. However, in [26] and [15], we constructed the p-adic height on elliptic curves at supersingular primes by using certain norm systems instead of the universalnorm subgroup. This enables us to control integral properties of the p-adic height and we could prove thep-adic Gross-Zagier formula for elliptic curves at supersingular primes in [15].

The aim of this paper is to generalize the norm construction of the p-adic height on elliptic curves in [26]and [15] to abelian varieties, and as an application we prove the p-adic Gross-Zagier formula for newformswith general coefficients (not necessarily in Q) of weight 2 for Γ0(N) at good non-ordinary primes.

Shinichi KobayashiMathematical Institute of Tohoku University6-3, Aoba, Aramaki, Aoba-ku, Sendai, Japane-mail: [email protected]

191

[email protected]

192 Shinichi Kobayashi

2 Norm subgroup of formal groups.

In this section, we review the universal norm subgroup and its variant in the non-ordinary case by recallinga brief history and their roles in Iwasawa theory.

Let Cp be the p-adic completion of the algebraic closure of Qp and let K be a finite extension of Qp inCp. We consider a ramified Zp-extension K∞ of K in Cp (e.g. the cyclotomic Zp-extension) with the Galoisgroup Γ ∼= Zp. Let Kn be the n-th layer in K∞, namely, the unique extension of K in K∞ of degree pn. Wedenote the integer ring of Kn (resp. Cp) by OKn (resp. OCp ). For an abelian variety A over K, the universalnorm subgroup of A(K) is defined by

NA(K) :=⋂n

NKn/K A(Kn)

where NKn/K is the norm from A(Kn) to A(K). The importance of the universal norm subgroup in Iwasawatheory is already observed by Mazur [Mar11], and in fact, the behavior of this local object determines the(conjectural) Zp[[Γ ]]-corank of the global Selmer group over the Zp-extension (if A and the Zp-extensionare defined over a global number field). Schneider [32] showed that if A has good reduction, then

rankZpA(K)/NA(K) = (dim A− r)[K : Qp]

where r is the p-rank of the reduction of A. In particular, NA(K) is of finite index in A(K) if and only if A hasordinary reduction. This fact is crucial for the norm construction of the p-adic height pairing by Schneider.

Now we slightly change the viewpoint. Instead of considering the universal norm subgroup, we considerthe group of norm compatible systems

lim←−n

A(Kn) := (Pn)n ∈∞

∏n=0

A(Kn) | TrKn+1/KnPn+1 = Pn.

Here we used the additive notation TrKn+1/Kn for the norm instead of NKn+1/Kn . The Iwasawa algebra Λ =Zp[[Γ ]] acts naturally on lim←−n

A(Kn) and by the result of Schneider, lim←−nA(Kn) is a finitely generated Λ -

module of free rank r[K : Qp]. In particular, lim←−nE(Kn) = 0 for an elliptic curve E with good supersingular

reduction.Nasybullin and Perrin-Riou showed that there is a variant of norm compatible systems which has suf-

ficient information also in the supersingular case. Now we assume K = Qp and K∞ is the cyclotomic Zp-extension. We fix a system of p-power roots of unity (ζpn)n such that ζ

ppn+1 = ζpn and ζp is a primitive p-th

root of unity. We denote the maximal ideal of Zp[ζpn+1 ] by mn and the maximal ideal of Zp[[T ]] by mT . Let∆ be the torsion part of Gal(Q(ζp∞)/Q) and let κ be the cyclotomic character which canonically identifiesΓ with 1+2pZp. The group Gal(Q(ζp∞)/Q) acts naturally on Zp[[T ]] by γ ·T = (T +1)κ(γ)−1. Let Lp bethe p-Euler factor polynomial X2−apX + p with ap = 1+ p− ]E(Fp). Perrin-Riou [26] showed that thereare sufficiently many “ Lp-norm systems"

lim←−Lp

E(Kn) := (Pn)n ∈∏n

E(Kn) | TrKn+2/KnPn+2−apTrKn+1/KnPn+1 + pPn = 0,

which also has a non-trivial subgroup

(Pn)n ∈∏n

E(Kn) | TrKn+2/Kn+1Pn+2−apPn+1 +Pn = 0. (1)

These norm systems play important roles in the Iwasawa theory at supersingular primes, for example,see [26], [14]. (Actually, the Russian mathematician Nasybullin had already noticed the importance ofsuch norm systems more than 10 years before Perrin-Riou and obtained the growth formula of the Tate-Shafarevich group at supersingular primes. See [21].) We recall that the norm compatible system for theformal multiplicative group Gm can be obtained as follows. For x ∈ Gm(Fp[[T ]])∆ , there is a canonical liftx ∈ Gm(Zp[[T ]])∆ of x fixed by the Coleman norm operator. Then x(ζpn+1 − 1) for n gives a norm com-patible system. Let E be the formal group of E. Perrin-Riou showed that for x ∈ E(Fp[[T ]])∆ , there is alsoa canonical lift x ∈ E(Zp[[T ]])∆ which satisfies “an Lp-relation" for a certain norm operator ψ related to

The p-adic height pairing on abelian varieties at non-ordinary primes 193

the Dieudonné module of E. Then x(ζpn+1 − 1) gives an Lp-norm system. We denote by ZE the image ofE(Fp[[T ]])∆ in E(Zp[[T ]])∆ by the lift. Then ZE is a Λ -module of rank 2 and we also have a Λ -submoduleZLE⊂ ZE of rank 1 related to the norm relation in (1). In [26], Perrin-Riou gave a new construction of the

p-adic height pairing on E by using these modules. (ZLEplays a role of the splitting of the Hodge filtration.)

For the calculation of the p-adic height of Heegner points, we need to characterize ZE and ZLEpurely in

terms of norm systems. In other words, we want to know which Lp-norm systems can be obtained from ZE .This is also a question proposed by Perrin-Riou before Lemme 4.8 of [26]. It is easy to see that the questionis equivalent to the problem of the interpolation of Lp-norm systems by power series, and the answer isgiven as a generalization of the theory of Coleman power series for E in [15].

The main theorem of the theory of Coleman power series says that any norm compatible system of Gmcan be interpolated by a power series, but in our case, there is a rather artificial element in lim←−Lp

E(Kn)

which cannot be obtained by elements in ZE (cf. [15, Remark 3.7]). Therefore we need to define a goodclass of norm systems in lim←−Lp

E(Kn), called admissible norm systems.

Definition 1. An Lp-norm compatible system (Pn)n ∈ lim←−LpE(mn) is called admissible if the p-th power

Frobenius map on E(OKn/p) sends Pn mod p to Pn−1 mod p for all n≥ 1. In other words, by the reductionmod p, the system (Pn)n defines a point in E(R) with Fontaine’s

R := (xn)n ∈∏(OCp/p) |xpn+1 = xn.

Then the Coleman power series theory can be generalized as follows.

Theorem 1 ([15]). Let (Pn)n be an Lp-norm compatible system. Then there exists a power series f ∈E(mT )

∆ such thatf (ζpn+1 −1) = Pn

if and only if (Pn)n is admissible. Furthermore, such f satisfies

(ψ2−apψ + p) logE f = 0 (2)

where logE is the formal logarithm of E and ψ is the Qp-linear trace operator for the Frobenius lift ϕ

characterized by ϕ(T ) = (T + 1)p− 1. Conversely, a power series in mT satisfying the relation (2) givesan admissible Lp-norm compatible system. In other words, the Λ -module ZE is isomorphic to the module ofadmissible Lp-norm compatible systems.

In [15], this theorem is proved in a similar way as in Washington’s book [34, Theorem 13.38], but itis much easier to use the theory of norm fields by Fontaine-Wintenberger as in the classical case. KazutoOta [24] also pointed out that Theorem 1 can be easily generalized to higher dimensional formal groups ofarbitrary finite height over absolutely unramified local rings by using the result of Knospe [13]. We recallthe result of Knospe-Ota.

First we specify the setting and fix some notation. Let K be the fractional field of the ring of Witt vectorsW =W (k) of the finite field k of q = pν elements. We denote the Frobenius of W by σ . We fix a uniformizerπ ∈ Zp and a Frobenius lift ϕ ∈W [[T ]] such that

ϕ(T )≡ πT mod deg2, ϕ(T )≡ T p mod pW [[T ]],

and consider the associated Lubin-Tate formal group Fπ of height 1 over Zp. We fix a system of π-powertorsion points (ϖn)n such that [π]ϖn+1 = ϖn where [π] is the multiplication by π of Fπ and ϖ1 is a non-zeroπ-torsion point. Let K∞ be the field obtained from K by adjoining all π-power torsion points of Fπ and Knthe field of πn-torsion points over K. (So now K∞ is not a Zp-extension but a Z×p -extension. However, thewhole story does not essentially change.) We denote the maximal ideal of the integer ring OKn by mn andthe maximal ideal of W [[T ]] by mT . Let ψ be the unique σ−1-semilinear map on W [[T ]] satisfying

ψ ϕ = p, ϕ ψ( f )(T ) = ∑[π]ϖ=0

f (T ⊕Fπϖ)

where the sum runs through all π-torsion points and ⊕Fπis the addition of Fπ .


Now we consider a d-dimensional formal group G = Spf W [[X1, . . . ,Xd ]] over W of finite height h. Forsimplicity, we use the multi-index notation and write the variables (Xi)i simply by X . (We always use theletter T as a one-variable parameter.) We let

PX = f (X) ∈ K[[X ]] | d f ∈ Ω1W [[X ]]/W , f (0) ∈ pW

and PX =PX/(pW [[X ]]). Similarly, we define one-variable PT and PT . Let F be the Frobenius on PX ,namely, the σ -semilinear ring morphism defined by X 7→ X p, and let V be the Verschiebung on PX definedas the σ−1-semilinear map ∑n anXn 7→ ∑n pσ−1(anp)Xn. The action of ϕ and ψ are extended on PX andthey are lifts of F and V on PX .

In our setting, the Dieudonné module MG of the special fiber of G may be defined explicitly as follows.For a closed form ω ∈ Ω1

W [[X ]]/W , we define the primitive function Fω of ω by the unique power series inPX such that dFω = ω and Fω(0) = 0. We put

Z1p(G ) := ω ∈ Ω

1W [[X ]]/W | dω = 0, Fω(X⊕GY )−Fω(X)−Fω(Y ) ∈ pW [[X ]],

B1p(G ) := d f ∈ Ω

1W [[X ]]/W | f (X) ∈ pW [[X ]].

Here ⊕G is the addition of G . We also consider the space of invariant differentials

LG := ω ∈ Ω1W [[X ]]/W | Fω(X⊕GY ) = Fω(X)+Fω(Y ).

Then we define the Dieudonné module MG by

MG := Z1p(G )/B1

p(G ),

which depends only on the special fiber G . The action of ϕ and ψ induces the Frobenius F and the Ver-schiebung V on MG , and MG is a free W -module of rank h. The Hodge filtration of MG is given by

FiliMG =

MG (i < 1),V LG (i = 1)0 (i > 1).

(We use the classical normalization of the Dieudonné module as in [9], [26]. The reader may consult [12]for various normalizations, especially §5.5. Our module is Dp(G/W (k)) there. The map p−1F defines theisomorphism from MG⊗Qp to the “usual" one by Grothendieck and Mazur-Messing. [12, (5.5.7)].) Let Lbe a finite extension of Qp. A splitting of the Hodge filtration of MG,L := MG⊗Zp L is an L-vector subspaceN ⊂MG,L such that MG,L = N +Fil1MG,L and N∩Fil1MG,L = 0.

We fix a basis ω1, . . . ,ωd of LG and denote their primitive functions by `1, . . . , `d . Then the formallogarithm `G is given by

`G : G →G⊕da , X 7→ (`1(X), . . . , `d(X)).

We consider a monic polynomial Q(t) = ∑mn=0 antn ∈ Zp[t] satisfying two conditions:

• Q(V )ωi = 0 for all i = 1, . . . ,d in MG .• all roots have p-adic absolute values strictly greater than |p|p = 1/p.

For example, the characteristic polynomial det( tν I−V ν |MG ⊗K ) as a K-vector space satisfies these con-ditions. If G is the basechange of a formal group G0 defined over Zp and ωi’s are invariant differentials ofG0 over Zp, we may take a smaller polynomial det( tI−V |MG0 ⊗Qp ) as Q(V ).

Definition 2. An element (Pn)n ∈∏∞n=1G

(n−1)(mn) is called a Q-norm system if it satisfies

trn+m/n `(n+m−1)G (Pn+m)+am−1trn+m−1/n `

(n+m−2)G (Pn+m−1)+ · · ·+a0`

(n−1)G (Pn) = 0

where we put G(n) := Gσ−n, `(n) := `σ−n

G and trn+m/n := trKn+m/Kn is the usual trace of fields. We denote theset of Q-norm systems by lim←−Q

G(n−1)(mn). Furthermore, if the p-th power Frobenius map G(n)(OCp/p)→


G(n−1)(OCp/p) sends Pn+1 mod p to Pn mod p for all n, the system is called admissible.1 We also say(Pn)n ∈Qp⊗∏

∞n=1G

(n−1)(mn) is admissible if (mPn)n ∈∏∞n=1G

(n−1)(mn) is admissible for some non-zerointeger m.

An important example of an admissible norm system is a system of Heegner points of higher order overring class fields of an imaginary quadratic field with a split prime p (so locally over p it defines a system ina height one Lubin-Tate tower).

We say that a system (Pn)n ∈Qp⊗∏∞n=1G

(n−1)(mn) is interpolated by an element f ∈ G(mT ) if

f (n−1)(ϖn) = Pn

for all n≥ 1.

Theorem 2 ([13], [24]). Let (Pn)n ∈ Qp⊗∏∞n=1G

(n)(mn) be a Q-norm system. Then (Pn)n is interpolatedby f ∈ G(mT ) if and only if (Pn)n is admissible. Such f satisfies Q(ψ)(`G f ) = 0. Conversely, a powerseries in mn satisfying this relation gives an admissible Q-norm system.

For p > 2, the theorem holds without ⊗Qp.As in the classical case, the proof of Theorem 2 is obtained by combining the injectivity of lim←−Q

G(n−1)(OKn)→lim←−Q

G(n−1)(OKn/p) and the following two isomorphisms. One is the isomorphism by the theory of norm

fields G(k[[T ]])∼= lim←−nG(n−1)(OKn/p) where the transition map is the p-th power Frobenius. The other is an

isomorphism obtained by the Perrin-Riou lift ([26, §4.1], [15, §3.2]) between G(k[[T ]]) and ZG, the Zp[[Γ ]]-submodule of G(mT ) consisting of elements f such that Q(ψ)(`G f ) = 0. We briefly recall the Perrin-Rioulift. By Fontaine-Honda, we have a canonical isomorphism of Zp-modules

G (k[[T ]]) = HomW [F,V ](MG ,PT ). (3)

Perrin-Riou [26, §4.1] (see also [24] for our Lubin-Tate case) constructed the canonical lifting

HomW [F,V ](MG ,PT ) −→ HomW (MG ,PT ), x 7−→ x (4)

characterized byψ(ϕ(x(m))− x(Fm)) = 0

for all m ∈ MG , or in other words, ψ(x(m)) = x(V m) for all m. She also showed that this condition isequivalent to

P(ψν/pν)(x(m)) = 0

for all m ∈MG where P is the characteristic polynomial

P(t) = detW ( I− tFν |MG ) ∈ Zp[t].

By Theorem 2, the morphism

G (k[[X ]])⊗Qp→Qp⊗∞

∏n=1

G(n−1)(mn), x 7→ (ω 7→ x(n−1)(ω)(ϖn)) (5)

is an isomorphism onto the set of admissible Q-norm systems. Here we identify G(n−1)(mn)⊗Qp with itstangent space HomW (LG(n−1) ,Kn). Since the Perrin-Riou lift does not depend on Q, the set of admissibleQ-norm systems is independent of the choice of Q. Therefore, we call the set of admissible Q-norm systemsjust the set of admissible norm systems for G .

Following Perrin-Riou, we consider the morphism of Zp-modules

δG : G (k[[X ]])→ HomW (M(1),W [[T ]]ψ=0)

defined by

1 The definition of the admissibility is different from that of [15], but they are equivalent by Theorem 2.


x 7−→ p−1(ϕ(x(m))− x(Fm)).

Here M(1) = M⊗σ W is the twist of M by σ . We write δG simply by δ if there is no fear of confusion. Weput u = π/p. Then the kernel and the cokernel of δG are given by

Ker δG∼= HomW (M(1),W )V=u logFπ

(T )

via (3) and (4), and

Coker δG∼= [HomW (M(1),W )/(V −u)HomW (M(1),W )](1). (6)

Here by definition, x ∈ HomW (M(1),W )V=u if and only if σ−1x(V−1(m)) = ux(m) for all m ∈ M(1), orequivalently, x(Fm) = πσx(m) for all m. The isomorphism (6) for the cokernel is given by looking at thefirst derivative, and the last symbol (1) in (6) is the twist by the Lubin-Tate character κπ for Fπ . SinceW [[T ]]ψ=0 is a free W [[Gal(K∞/K)]]-module of rank 1 by the natural action of Gal(K∞/K) via κπ , we havethe following.

Proposition 1. Suppose that Ker δG =Coker δG = 0 (e.g. G is biconnected), then G (k[[X ]]) has a W-modulestructure and it is a free W [[Gal(K∞/K)]]-module of rank h. Hence so does the module of admissible Q-normsystems of G.

So far, we saw the theory of the universal norm subgroup and its variant only for formal groups. Herewe add a few comments on the case of Galois representations. The concept of the universal norm subgroupis generalized in terms of Bloch-Kato’s local Selmer groups, and investigated in [30] mainly for crystallinerepresentations and [2] for de Rham representations. A generalization of Q-norm compatible systems forcrystalline representations is obtained from discussions in §2.3 and §2.4 of [28] (especially from Proposition2.4.2), which is the key of the construction of Perrin-Riou’s big exponential map. Then the Perrin-Riou mapis generalized for de Rham representations by Colmez [8] and for de Rham (ϕ,Γ )-modules by Nakamura[20].

3 Norm construction of the p-adic height pairings on abelian varieties atnon-ordinary primes

The p-adic height pairing on abelian varieties is constructed by P. Schneider [31] at ordinary primes byusing the universal norm subgroup. Zarhin [36] gave a different construction including the non-ordinarycase. His observation is that the p-adic height pairing is constructed depending on a choice of a splitting ofthe Hodge filtration of the p-adic de Rham cohomology as Qp-vector spaces (not as K-vector spaces). Inthe good ordinary case, if we take the splitting obtained by the unit root subspace of the Dieudonné module,the resulting height pairing coincides with the pairing by Schneider. In this section, we use G(k[[T ]]) orequivalently, admissible norm systems to construct the p-adic height pairing at non-ordinary primes insteadof the universal norm subgroup. We define a certain class of Qp-vector subspaces of the Dieudonné module,called adequate. We associate an adequate subspace to a Zp[[Γ ]]-submodule of G(k[[T ]]) that plays a similarrole with the splitting of the Hodge filtration in Zarhin’s construction, and our height pairing depends onthe choice of the adequate subspace. We show a correspondence between adequate subspaces and splittingsof the Hodge filtration, and compare our height with Zarhin’s. The correspondence is essentially given asthe p-adic interpolation (smoothing) factor appeared in the Perrin-Riou map and this also suggests that thenorm construction is more suitable for investigating integral properties.

3.1 Submodules corresponding to splittings of the Hodge filtration

In this subsection, we construct a class of Zp[[Γ ]]-submodules of G(k[[T ]]) that corresponds to splittingsof the Hodge filtration as Qp-vector spaces. As pointed out in [22, §2.9], we require only the splitting asQp-vector spaces but not as K-vector spaces. Our correspondence between Zp[[Γ ]]-submodules of G(k[[T ]])and splittings of the Hodge filtrations does not respect the structure of W -modules.


We follow the setting and notations in Section 2 if not otherwise specified. We denote by t the usualtrace K→ Qp and we also use the same symbol for maps induced by t by abuse of notation. For example,it induces canonical isomorphisms of Zp-modules

t : HomW (M(1)G ,W [[T ]]ψ=0)∼= HomZp(MG,Zp[[T ]]ψ=0),

t : HomW (M(1)G ,K)∼= HomZp(MG,Qp).

Let L be a finite extension of Qp in Cp. We often abbreviate X⊗Zp L by XL for a Zp-module X .

Definition 3. For an L-vector subspace N ⊂MG,L, we define ZG,N(k[[T ]]) as the inverse image by tδG inG(k[[T ]])L of HomL(MG,L/N,Zp[[T ]]

ψ=0L ), namely,

ZG,N(k[[T ]]) := x ∈ G(k[[T ]])L | t (ϕ x(ω)− x(Fω)) = 0 for all ω ∈ N.

Remark 1. If N is a K⊗L-submodule of MG,L, the above condition is equivalent to ϕ x(ω) = x(Fω) for allω ∈ N since the pairing K×K→ Qp,(x,y) 7→ t(xy) is non-degenerate. Therefore this definition coincideswith that in [26] and [15].

By the morphism (5), the module ZG,N(k[[T ]]) corresponds to a Λ -submodule of rank [K :Qp]dimL(MG,L/N)of admissible norm systems for G. It seems not easy to describe ZG,N(k[[T ]]) purely in terms of norm re-lations for a general N but at the end of this subsection we give such descriptions for special types of N,which is sufficient for our application.

The mapZp[[T ]]ψ=0→ Zp, f 7→ tr1/0 f (ϖ1)

induces an isomorphism

HomL(M(1)G,L,Zp[[T ]]

ψ=0L )/(γ−1)

∼= HomL(MG,L,L).

Here, for simplicity, we write X/(γ − 1)X as X/(γ−1) for a Zp[[Γ ]]-module X . Combined with t δG, this

induces a map δG : G(k[[T ]])L/(γ−1)→ HomL(MG,L,L),

ω 7−→ p−1t tr1/0[(ϕ x(ω)− x(Fω))(ϖ1)] = p−1t x((F−1)ω)(0). (1)

Here we used the following easy lemma.

Lemma 1. Let f (T ) be an element of PT . Then

ψ( f σ )(ϖn) = trn+1/n( f (ϖn+1))

for n≥ 1 andψ( f σ )(0) = tr1/0( f (ϖ1))+ f (0).

Proposition 2. The morphism δG is an isomorphism. In particular, the restriction of δG induces an isomor-phism

δG,N(= δN) : ZG,N(k[[T ]])/(γ−1)∼= HomL(MG,L/N,L). (2)

Proof. The proof follows from the description of the kernel and the cokernel of δG.

Corollary 1. The morphism

G(k[[T ]])L/(γ−1)→ HomW (MG,K)L, x 7→ (ω 7→ x(ω)(0))

is an isomorphism.

Proof. This follows from Proposition 2 and (1). Note that F−1 is invertible since G is connected. See alsoClaim 2 in p.256 of [13].


For i≥ 1, we consider the composition

ΣG,i : G(k[[T ]])⊗Qp → Qp⊗ lim←−Q

G(n−1)(mn) → G(i−1)(mi)⊗Qp

with (5). We also put ΣG := Tr1/0ΣG,1. We omit the index G in ΣG,i and ΣG if there is no fear of confusion.We sometimes identify an element of G(k[[T ]])⊗Qp with the admissible norm system by (5), and then ΣG,iis just the projection map (Pn)n 7→ Pi and ΣG is (Pn)n 7→ Tr1/0P1.

Proposition 3. The morphism ΣG,i is surjective. If p > 2, it is surjective without the tensor ⊗Qp.

Proof. First consider the case p > 2. Suppose that Pi ∈ G(i−1)(mi) is given. Then Pi := Pi mod p ∈G(i−1)(OKi/p) can be extended to a family

(Pn)n ∈ lim←−x 7→xp

G(n−1)(OKn/p) = G(k[[T ]]).

Hence by Perrin-Riou’s lift, we have an admissible norm system (Qn)n such that Qi ≡ Pi mod p. ByNakayama’s lemma, we have the conclusion. The case p = 2 is similar. (Note that in general the Perrin-Riou lift (hence Σi) is well-defined only after the tensor Qp when p = 2. cf. [15, Proposition 3.1].)

By the logarithm map, G(W )⊗Qp is naturally identified with HomW (Fil1M(1)G ,K) = HomW (V L(1)

G ,K).Then Σ is regarded as the morphism of L-vector spaces

G(k[[T ]])L→ HomW (V L(1)G ,K)L, x 7−→ (V ω 7→ tr1/0x(ω)(ϖ1)).

This map is extended as

Σe : G(k[[T ]])L→ HomW (M(1)

G ,K)L, x 7−→ (ω 7→ p−1tr1/0x(Fω)(ϖ1)).

Proposition 4. Suppose that the eigenvalues of V on MG as Qp-vector spaces are different from 1. Then Σ e

is surjective. In particular, Σ is also surjective.

Proof. By Lemma 1, we have

t tr1/0x(ω)(ϖ1) = t x((V −1)ω)(0). (3)

Since V −1 is invertible on MG by our assumption, the assertion follows from Corollary 1.

The map Σ(= ΣG) induces an L-linear map ΣN(= ΣG,N) : ZG,N(k[[T ]])→G(pW )L, which factors through

ΣN(= ΣG,N) : ZG,N(k[[T ]])/(γ−1) −→ G(pW )L.

Similarly, Σ e induces the morphism

ΣeN : ZG,N(k[[T ]])→ HomW (M(1)

G ,K)L

andΣ

eN : ZG,N(k[[T ]])/(γ−1) −→ HomW (M(1)

G ,K)L.

Suppose that the Verschiebung V on MG as a Qp-linear map does not have 1 as an eigenvalue. Then byProposition 2 and Proposition 4, we have

t Σe : G(k[[T ]])/(γ−1)

∼= HomZp(MG,Qp)

and hence the Qp-linear isomorphism

Φ = (t Σ e δ−1G )∨ : MG

∼=−−−−→ MG

where ∨ is the dual as Qp-vector spaces.By (1) and (3), we have the following.


Proposition 5. The Qp-linear map Φ is given by

MG −→ MG, ω 7−→ (p−F)(F−1)−1ω.

Note that Φ is the interpolation factor of Perrin-Riou’s theory. (cf. [29], [8, Théorèm 3 and Remarque 5(iii)]. It looks slightly different because of our classical normalization of Dieudonné modules.) The map Φ

is extended on MG,L linearly.

Corollary 2. Assume that 1 is not an eigenvalue of V on MG⊗Qp as a Qp-vector space. Let N be anL[F ]-submodule of MG,L. Then Φ(N) = N.

We determine Qp-vector subspaces that are the image by Φ of complimentary subspaces to the Hodgefiltration of MG,L.

Definition 4. We say an L-vector subspace N⊂MG,L is adequate if it satisfies the following two conditions.

1. dimL (MG,L/N) = [K : Qp]dimG.

2. ΣN is surjective.

Since dimL ZG,N(k[[T ]])/(γ−1) = dimL(MG,L/N) by Proposition 2, the adequateness is equivalent to re-quiring that

t ΣN : ZG,N(k[[T ]])/(γ−1) −→ HomL(Fil1MG,L)

is an isomorphism of L-vector spaces. Therefore if N is adequate, Σ−1N and Σ induce an L-linear map

HomL(Fil1M(1)G,L,L)

∼= ZG,N(k[[T ]])/(γ−1)→ G(k[[T ]])/(γ−1)→ HomL(MG,L,L),

which is the splitting of the Hodge filtration by Φ−1(N). Conversely, a splitting of the Hodge filtrationdefines an adequate subspace by reversing the argument. Hence we have

Proposition 6. Assume that 1 is not an eigenvalue of V on MG⊗Qp as a Qp-vector space. Then Φ definesa one-to-one correspondence between splittings of Hodge filtrations as L-vector spaces and adequate L-vector subspaces of MG,L.

This correspondence does not respect the canonical splitting by Wintenberger [35] (cf. [15, Remark 4.7])but it respects splittings as L[F ]-modules by Corollary 2.

Now we study admissible norm systems coming from a particular type of N. Let e be an idempotent ofEnd(G /W )⊗L. Let q be an L-vector subspace of the polynomial ring L[t] and consider an L-vector subspaceN of MG,L satisfying

eN = qeFil1MG,L := f (V )ω | f ∈ q, ω ∈ eFil1MG,L. (4)

For a polynomial f (t) = aktk +ak−1tk−1 + · · ·+a0 ∈ L[t], we define a map

f (tr) :∞

∏n=1

G(n−1)(mn) −→∞

∏n=1

K⊕dn , (Pn)n 7−→ (qn)n

whereqn := aktrn+k/n `

(n+k−1)G (Pn+k)+ak−1trn+k−1/n `

(n+k−2)G (Pn+k−1)+ · · ·+a0`

(n−1)G (Pn).

By definition, the kernel of f (tr) is the set of f -norm systems.

Proposition 7. Suppose that an L-vector subspace N⊂MG,L satisfies (4) and let (Pn)n ∈L⊗∏∞n=1G

(n−1)(mn)be an admissible norm system. Then (ePn)n is an admissible norm system in the image of ZG,N(k[[T ]]) by(5) if and only if (ePn)n is in the kernel of

pr f (tr) : L⊗∞

∏n=1

G(n−1)(mn) −→ L⊗∞

∏n=2

(Kn/Kn−1)⊕d

for all f ∈ q, namely,


eqn = ak trn+k/n `(n+k−1)G (ePn+k)+ · · ·+a0 `

(n−1)G (ePn) ∈ K⊕d

n−1.

Here pr is the natural projection.

Proof. Since (Pn)n is an admissible norm system, it is the image of x ∈ G(k[[T ]])L by (5). By definition wehave ex ∈ ZG,N(k[[T ]]) if and only if

(ϕ ψ− p) f (ψ)ex(ω)(T ) = ϕ x(η)(T )− x(Fη)(T ) = 0

where η = e f (V )V ω ∈ eN for any f ∈ q and V ω ∈V LG = Fil1MG . (Note that ex = x e and ψ x = xV ,and eN is a K-vector space by (4).) If this holds, then trn/n−1eqn = peqn for n≥ 2 by Lemma 1. In particular,eqn ∈ K⊕d

n−1. Conversely, if (ePn)n is in the kernel of pr f (tr), then

(ϕ ψ− p) f (ψ)ex(ω)(T )|T=ϖn = 0

for V ω ∈ Fil1MG and n≥ 2. Hence

(ϕ ψ− p) f (ψ)ex(ω)(T ) = clogFπT

for some constant c. By applying ψ , we have c = 0.

Now we assume that eN = qeFil1MG,L is an F-invariant subspace of MG,L. Let Q ∈ L[t] be such thatQ(V )Fil1MG,L = 0 and all roots of Q have p-adic absolute values strictly greater than |p|p = 1/p.A@Then eN is written in the form eN = f eFil1MG,L with f |Q.

Proposition 8. Let N be a L-vector subspace of MG,L such that eN = f eFil1MG,L and eN is F-invariant.Let (Pn)n be an admissible norm system for G(k[[T ]])L. Then (ePn)n is an admissible norm system forZG,N(k[[T ]]) if and only if (ePn)n is an f -norm system.

Proof. Suppose that (Pn)n is the image of x ∈ G(k[[T ]])L by (5). We take an element g of the ideal ( f ) ⊂Zp[t]. Then egFil1MG,L ⊂ eN by our assumption, and as in the proof of Proposition 7, we suppose that

(ϕ ψ− p)g(ψ)ex(ω)(T ) = 0

for ω ∈ LG . By putting T = ϖn for n≥ 0, we have g(ψ)ex(ω)(ϖn) ∈ K⊕dn−1 for n≥ 2 and

g(ψ)ex(ω)(ϖ1) = g(ψ)ex(ω)(0)

by Lemma 1. Then by taking g = tm f (m = 0, . . . ,n), we have

f (ψ)ex(ω)(ϖn) = f (ψ)ex(ω)(0).

Hence we have eqn ∈ K⊕d and it is constant for varying n. However, eqn is an H-norm system for thepolynomial H = Q/ f with H(p) 6= 0. Thus eqn = 0. The converse is proven similarly as in the proof ofProposition 7.

Example 1. Let E be the formal group of an elliptic curve over Qp with good supersingular reduction. Firstwe let N = LE ⊗Qp. Then N = (V −ap)Fil1ME,Qp

, and by Proposition 7, an admissible norm system (Pn)n

comes from ZE,N(k[[T ]]) if and only if

Trn+1/nPn+1−apPn ∈ E(mn−1) (5)

for n ≥ 2. Then combined with Trn+1/n−1Pn+1−apTrn/n−1Pn + pPn−1 = 0, we have Trn+1/nPn+1−apPn +

Pn−1 = 0. Conversely, this relation implies (5). Next we suppose N = (V −α)Fil1ME,L = L(F − β )ωEwhere α,β are the roots of t2−apt + p = 0 and L =Qp(α). Then N is the α-eigenspace of the Frobenius.Hence Proposition 8 implies that an admissible norm system (Pn)n comes from ZE,N(k[[T ]]) if and only ifTrn+1/nPn+1 = αPn for all n≥ 1.


3.2 The Zarhin-Nekovár construction of p-adic height pairings

In this subsection, we recall Zarhin’s construction of the p-adic height pairing on abelian varieties, which isgeneralized to Galois representations by Nekovár.

Let F be a finite extension of Q in a fixed algebraic closure Q. We fix an embedding ιp : Q → Cp. Weassume that F is unramified over p. Let L be a finite extension of Qp (coefficient field) as before. Let logpbe the p-adic logarithm on Z×p such that logp p = 0. We define the cyclotomic logarithm `F,v on F×v at anon-archimedean place v by

`F,v(x) =

− logp |x|v = v(x) logp N(v) if v - p− logp NFv/Qp(x) if v | p

where N(v) is the number of elements of the residue field of F at v and we normalize as v(π) = 1 for auniformizer π at v of F . We also denote `F,v by `v if there is no fear of confusion. We define the globalcyclotomic logarithm `F by `F := ∑v `F,v. Then `F(x) = 0 for x ∈ F×.

Let A be an abelian variety over F and A∨ the dual abelian variety of A. For a finite place v of F , letD0(A)(Fv) be the group of divisors algebraically equivalent to 0 defined over Fv and Z0(A)0(Fv) the groupof zero cycles of degree 0 defined over Fv. We denote by

(D0(A)(Fv)×Z0(A)0(Fv))e

the subgroup of D0(A)(Fv)×Z0(A)0(Fv) consisting of pairs with disjoint support.Let N be an L-vector subspace of the Dieudonné module MA⊗L complementary to the Hodge filtration.

We recall the local p-adic height pairing

〈 , 〉p,v,N : (D0(A)(Fv)×Z0(A)0(Fv))e −→ L

associated to N with the logarithm `v : F×v →Qp.First we recall the theta group. (See [17], [19].) Let A/OFv be the Néron model of A/Fv and A0 the

identity component. Let A ∨ be the Néron model of A∨. Then the rational equivalence class D of D ∈D0(A)(Fv) defines a point in A∨(Fv) = A ∨(OFv) = Ext1f pp f (A

0,Gm). Hence we have an exact sequence

1 −−−−→ Gm −−−−→ XD −−−−→ A 0 −−−−→ 1 (1)

as abelian fppf-sheaves on OFv (actually, it is exact as Zariski sheaves), and XD is represented by a smoothseparated commutative group scheme over OFv .

Over SpecFv, this exact sequence is isomorphic to

1 −−−−→ Gm −−−−→ XD −−−−→ A −−−−→ 1 (2)

where XD is given by Spec(SymOA(−D))\the zero section, the line bundle associated to OA(D) minusthe zero section with group law from the primitivity (algebraically equivalent to zero). Hence attached to D,there is a section sD : A\ |D| → XD which is canonical up to a translation by an element of Gm. Note that theisomorphism class of the extension (2) corresponds to the rational equivalence class of D. However, here,we choose the particular choice of the extension in the isomorphism class with the section sD associated tothe divisor D. We identify XD⊗Fv with XD.

Suppose that A has good reduction at all places over p. We now define a morphism `D,v that makes thefollowing diagram commutative.

0 // O×Fv⊗L //

XD(OFv)⊗L //

A 0(OFv)⊗L // 0

0 // F×v ⊗L //

`v

XD(Fv)⊗L //

`D,v

A(Fv)⊗L // 0

L L.


(The rows are exact but vertical sequences are not.) When v - p, the logarithm `v is unramified at v, namely,`v is trivial on O×Fv

. Then `v is uniquely extended to `D,v so that the restriction to XD(OFv) is trivial. Nowassume that v | p. Let MA be the Dieudonné module of the special fiber of A at v. We take an L-vectorsubspace N of MA ,L such that MA ,L = N⊕ Fil1MA ,L as L-vector spaces, which defines a splitting as L-vector spaces of the exact sequence

0 // Fil1MA ,L // MA ,L // MA ,L/Fil1MA ,L // 0.

Since MV=1A ,L = 0, the exact sequence

0 // MA ,L // MXD,L// MGm,L

// 0.

splits as F-modules by using the subspace MV=1XD,L⊗Qp Fv ∼= MGm,L. We put ND := N⊕ (MV=1

XD,L⊗Fv). Thenwe obtain an L-linear map

Fil1MXD,L→MXD,L/ND ∼= MA ,L/N ∼= Fil1MA ,L.

The pullback by this map defines a splitting as L-linear vector spaces

sD,v,N : A (Fv)⊗L = HomL(Fil1MA ,L,L) −→ HomL(Fil1MXD,L,L) = XD(Fv)⊗L.

Here for Y = A or XD, we identify

HomFv⊗L(Fil1M(1)Y ,L,Fv⊗L) = HomL(Fil1MY ,L,L)

by the trace map t : Fv→Qp. We extend the logarithm `v to `D,v on XD(Fv)⊗ZpL so that it is trivial on theimage of sD,v,N .

The local p-adic height pairing at a finite place v is defined as

〈D,a〉p,v,N := `D,v

(∏

isD(Pi)

ni

)

for a zero cycle a= ∑i ni(Pi) ∈ Z0(A)0(Fv) prime to D fixed by Gal(Fv/Fv). Here we are writing the grouplaw of XD multiplicatively. There is an ambiguity of a constant multiple for sD but the pairing is well-defined since ∑i ni = 0.

Proposition 9. The local p-adic height pairing

〈 , 〉p,v,N : (D0(A)(Fv)×Z0(A)0(Fv))e −→ L

has the following properties.

1. 〈 , 〉p,v,N is bilinear (whenever this makes sense).2. If D = (h) is principal, then

〈(h),a〉p,v,N = `v(∏i

h(Pi)ni)

for a zero cycle a= ∑ni(Pi) ∈ Z0(A)0(Fv) prime to D fixed by Gal(Fv/Fv).3. For any finite morphism φ : A→ A, we have

〈φ ∗D,a〉p,v,N = 〈D,φ∗a〉p,v,N .

4. For any divisor D ∈D0(A)(Fv) and any point x0 ∈ A(Fv)\ |D|, the morphism on A(Fv)\ |D|

x 7→ 〈D,(x)− (x0)〉p,v,N

is continuous for the v-adic topology on A(Fv) and the p-adic topology on L.

If v is prime to p, the pairing is characterized by the above properties.


The global p-adic height pairing can be defined as the sum of local p-adic height pairings

A∨(F)×A(F)→ L, (a,b) 7→∑v〈a′,b′〉p,v,N .

Here a′ is a divisor that represents a and b′ is a zero cycle ∑ni[bi] of degree zero with ∑nibi = b, and wechoose a′ and b′ so that they have no point in common. The value ∑v〈a′,b′〉p,v,N is independent of the choicea′ and b′.

3.3 Norm construction of p-adic height pairings

In this subsection, we give a norm construction of the p-adic local height pairing associated to an adequateL-vector subspace N of M Â ,L for the formal group A at v|p. Then we show that the pairing coincideswith that of Zarhin-Nekovár associated to the splitting of the Hodge filtration by c−1(Φ−1(N)) wherec : MA ,L→M Â ,L is the natural map induced by the formal completion. Note that if A is ordinary at v, N

must be zero by the adequateness, and the splitting coincides with that by the unit root subspace.

We use the same notation as in the previous subsection. We assume that v|p and put K = Fv and W =OFv

for simplicity. Let A be the formal group of A over W . The formal completion induces an exact sequenceof Dieudonné modules

0 // MunitA ,L

// MA ,Lc // M Â ,L

// 0.

where MunitA ,L is the unit root subspace of MA ,L of the Frobenius F . (cf. [12, Corollary 5.7.7, 5.7.8]). Let N

be an adequate L-vector subspace of MA,L. In the following we define a splitting

snormD,v,N : Â (Fv)⊗L −→ XD(Fv)⊗L.

Then the p-adic (local) height pairing is constructed just as Zarhin-Nekovár’s except that we use the localsection snorm

D,v,N instead of sD,v,N .We consider L-linear maps

ΣA,N : ZA,N(k[[T ]])/(γ−1) −→∼= A(pW )L, (1)

ΣXD,ND: ZXD,ND

(k[[T ]])/(γ−1) −→ XD(pW )L. (2)

Here we put ND := N⊕ (MV=1XD,L

⊗K). Note that ΣA,N is an isomorphism since N is adequate. On theother hand, by Proposition 2, we have isomorphisms

δA,N : ZA,N(k[[T ]])/(γ−1)∼= HomL(MA,L/N, L), (3)

δXD,ND: ZXD,ND

(k[[T ]])/(γ−1)∼= HomL(MXD,L

/ND, L). (4)

Since MXD,L= MA⊕(MV=1

XD,L⊗K) as L-vector spaces, isomorphisms (3) and (4) induce an isomorphism

ZA,N(k[[T ]])/(γ−1)∼= ZXD,ND

(k[[T ]])/(γ−1). (5)

Hence combined with (1) and (2), we have the desired section

snormD,N : Â (Fv)⊗L −→ XD(Fv)⊗L.

By construction, it is straightforward to see that snormD,N coincides with sD,N for

N = c−1(Φ−1(N)) = Φ−1(N)⊕Munit

A ,L


where we regard as MA ,L =MunitA ,L⊕M Â ,L by the splitting as F-modules. In particular, snorm

D,N does not dependon the choice of the Lubin-Tate extension of Qp.

Now we describe snormD,N in terms of admissible norm systems. Let N be an adequate L-vector subspace

of M Â ,L. For x0 ∈ Â (W )⊗L, we can take an admissible norm system (xn)n ∈ L⊗∏∞n=1 A

(n−1)(mn) inter-polated by an element of Z Â ,N(k[[T ]]) with Tr1/0 x1 = x0 by (1) (here we used the adequateness).

Lemma 2. There exists an admissible norm system (xn)n≥1 ∈ L⊗∏∞n=1 X

(n−1)D (mn) that is a lift of (xn)n for

the exact sequence

0 // Gm(mn)L // X (n−1)D (mn)L // Â (n−1)(mn)L // 0.

If (xn)n is a Q-norm system, then we can take (xn)n so that it is also a Q-norm system.

Proof. We take x∈ ZA,N(k[[T ]]) interpolating (xn)n. By (5), we can take a lift x∈ ZXD,ND(k[[T ]]) of x. Then

xn := ΣXD,n,ND(x) gives a desired system. Let R ∈ Zp[t] be a monic polynomial such that R(V )MA = 0

and all roots of R have p-adic absolute values strictly greater than |p|p = 1/p. Moreover, since A is theformal group of an abelian variety, we may choose R so that R(1) 6= 0. Then by the definition of ND, wehave R(ψ) ˆx(ω)(T ) = c(ω) logFπ

(T ) for some constant c(ω) for all ω ∈MXD. Hence (xn)n is an R-norm

system. Suppose that (xn) is a Q-norm system for Q∈Zp[t]. Then we may assume Q is a divisor of R and putR = PQ. We let yn = Q(tr)(xn). Then (yn)n is a P-norm system of Gm, but it must be also a norm compatiblebecause it is obtained by the Perrin-Riou lift. However, since P(1) 6= 0, this system must be trivial.

It is straightforward to see that the local section snormD,w,N can be given by

Â (W )L −→ XD(W )L, x 7−→ Tr1/0 x1.

Here we take an admissible Q-norm system (xn)n with Q(1) 6= 0 and Tr1/0 = x, then (xn) is an admissibleQ-norm lift of (xn)n.

We may also define a section

Â (W )L −→ XD(W )L/(Ker ˆw⊗L), x 7−→ Tr1/0 x1

for any admissible lift (xn). The well-definedness is checked as follows. If (yn)n is another admissible liftof (xn), then yn− xn defines an admissible norm system of Gm, which must be norm compatible. HenceTr1/0(y1− x1) is a universal norm in Gm(W ), and therefore contained in Ker ˆw⊗L.

4 An application for the p-adic Gross-Zagier formula

In this section we generalize the p-adic Gross-Zagier formula in [15] to newforms for Γ0(N) of weight 2with arbitrary Fourier coefficients (not necessarily in Q). Most parts of the proof in [15] also work in thiscase (or have been already proven in [15]). The missing part is the theory of the p-adic height pairing onabelian varieties at non-ordinary primes developed in this paper.

Let f = ∑∞n=1 an( f )qn be a normalized eigen newform for Γ0(N) of weight 2. Let p be a prime number

such that (p,N) = 1 and we fix a complex embedding ι∞ : Q → C and a p-adic embedding ιp : Q → Cp.We fix real and purely imaginary periods Ω

±f (e.g. Shimura periods) and a system of p-power roots of unity

(ζpn)n that is a generator of Zp(1). Let α be a root of t2−ap( f )t + p = 0 in Cp such that |p/α|p < 1 whereap( f ) is regarded as an element of Cp by ι∞ and ιp. It is known that there is a one-variable p-adic analyticdistribution dµ f ,α of order < 1 satisfying that∫

Z×pχ(x)dµ f ,α(x) =

τ(χ)

αnL( f ,χ,1)

Ωχ(−1)f

for a non-trivial Dirichlet character χ of conductor pn with the Gauss sum τ(χ), and


∫Z×p

dµ f ,α =

(1− 1

α

)2 L( f ,1)Ω

+f

.

(cf. [1], [18], [33].) Here the L-values on the right hand side are algebraic and regarded as in Cp by ι∞ andιp. Then the p-adic L-function Lp( f ,α,s) is defined as an analytic function of s on Zp by

Lp( f ,α,s) :=∫Z×p〈x〉s−1 dµ f ,α(x),

where 〈〉 is the natural projection Z×p → 1+2pZp.Let K be an imaginary quadratic field with discriminant dK =−δ such that (dK ,N p) = 1. Let ε be the

quadratic character attached to K /Q. We define the p-adic L-function of f over K by

Lp( f/K ,α,s) := Lp( f ,α,s)Lp( f ⊗ ε,ε(p)α,s)Ω

+f Ω

+f⊗ε

δ12

Ω f

where f ⊗ ε := ∑∞n=1 ε(n)anqn is the twist of f by ε and

Ω f :=∫

X0(N)(C)ω f ∧ iω f

for ω f = ∑anqndq/q = 2πi f (τ)dτ with q = exp(2πiτ). If all rational primes dividing N split in K , thenLp( f/K ,α,1) = 0 since L( f/K,1) = L( f ,1)L( f ⊗ε,1) = 0 by looking at the sign of the functional equa-tion.

Let X0(N) be the modular curve over Q for Γ0(N) and let J = J0(N) be its Jacobian variety. There is anidempotent e f in EndQ J⊗Q f such that

e f (Γ (X0(N),Ω 1X0(N)/Q)⊗Q f ) =Q f ω f .

Here Q f is the field ι−1∞ (Q(an( f )n)), and we let ω f be defined over Q f by ι∞. Let L be the p-adic closure

of ιp(Q f )(α) in Cp. Let MJ be the Dieudonné module attached to the special fiber of J at p. By the Albanesemap X0(N)→ J0(N), x 7→ [x]− [∞], the space of invariant holomorphic forms can be regarded as

LJ := Γ (X0(N),Ω 1X0(N)/Q)⊗QQp ⊂MJ⊗Zp Qp

and Fil1MJ⊗Qp =V LJ . Then the Frobenius F acts on the space e f (MJ⊗L) and the characteristic polyno-mial is t2− ap( f )t + p. Let Nα be a subspace of MJ ⊗L complementary to the Hodge filtration such thate f Nα is the α-eigensubspace of F of e f (MJ⊗L). Then we consider the p-adic height pairing on J0(N)(K )attached to Nα ⊗Q K (with the cyclotomic logarithm `K )

〈 , 〉p,K ,α : J0(N)(K )× J0(N)(K ) −→ L

where we identify J0(N) with its dual J0(N)∨ by the principal polarization.Let H be the Hilbert class field of K . We assume that all rational prime numbers dividing N split in K .

Then there exists a Heegner point zH ∈ X0(N)(H ) that represents a cyclic isogeny C/OK → C/N −1 oforder N where (N) =N N ∗ in OK . (There exist several choices of Heegner points but we fix one since thep-adic height does not depend on the choice.) We also put zH ∈ J0(N)(H ) by the negative of the Albanesemap (see Remark 4.2 in [15] for the sign convention) and put zK , f := e f TrH /K zH ∈ J0(N)(K )⊗Q f .

Then the p-adic Gross-Zagier formula for f is stated as follows.

Theorem 3. Suppose that p splits in K . Then

dds

Lp( f/K ,α,s) |s=1 = u−2(

1− 1α

)2(1− 1

ε(p)α

)2

〈zK , f ,zK , f 〉p,K ,α

where u = ]O×K /2.


The formula is proven in [25] for an ordinary p and in [15] for a supersingular p for f with rationalcoefficients. If the p-adic height is a non-trivial function, we can also show the same formula for an inertprime p by a simple trick. (See the proof of Theorem 5.8 in [15].) The strategy of the proof is as follows.We construct two p-adic modular forms F and G related to the p-adic height of the Heegner point andthe derivative of the p-adic L-function respectively. We compare their Fourier coefficients by independentcalculations and show that “the f -part" of F and G coincide, which gives the p-adic Gross-Zagier formula.

As in the complex case, the p-adic modular form F is defined as

F := ∑σ∈Gal(H /K )

∞

∑m=1〈zH ,Tmzσ

H 〉p,H ,α qm.

For G, we first construct the p-adic measure Φ on Z×p with values in the space of p-adic modular formsby the p-adic convolution of the Eisenstein and theta measure. Then put

G(s) :=∫Z×p〈x〉s−1dΦ

and G=(d/ds)G(s)|s=1. “The f -part" of G(s) is expected to be related to the p-adic L-function Lp( f/K ,α,s).In fact, if p is ordinary, this is done by using Hida’s ordinary projection which is considered as an analogueof Sturm’s holomorphic projection and also enables us to take the f -part of G(s) p-adically (essentially thePetersson inner product with f ). In the non-ordinary case, the convergence condition arising from taking thef -part is not admissible and the direct analogue to the ordinary case does not work. (In fact, Lp( f/K ,α,s)is in some sense a critical slope p-adic L-function since it cannot be characterized by the interpolationproperty because the corresponding power series has denominators of the logarithmic order.) In [15], toovercome this difficulty, we introduced a (naive) two variable p-adic Eisenstein measure Φ (2) on Z×p ×Z×pwhose restriction to the diagonal direction coincides with Φ (both variables are in the cyclotomic directionsand “the f -part" of the integral of the function 〈x〉s−1〈y〉t−1 with respect to Φ (2) corresponds to the productLp( f ,α,s)Lp( f ⊗ ε,ε(p)α, t)). This two-variable measure is constructed by using products of Eisensteinseries (essentially, Kato’s zeta element in the space of modular forms [11]). Then the convergence conditionrelated to taking the f -part of Φ (2) on the vertical and horizontal directions is good enough and we canapproximate the diagonal direction by using these directions to relate G(s) to Lp( f/K ,α,s). Calculationsconcerning G are written in [15] for modular forms of weight 2 for Γ0(N) with general coefficients.

For the calculation of Fourier coefficients of F , we use the decomposition of the global height into thelocal heights. The local height at v - p is described geometrically as the intersection number on the integralmodel of X0(N). Hence the calculation is reduced to the classical case [10]. For the local height at v | p,we show that it is essentially equal to zero. (We show that the “ f -part " vanishes. 2). The proof in [15] alsoworks in our case if we use the theory of the p-adic height developed in this paper. So we do not recall thedetails, but we just recall the setting to apply to our theory.

We now assume that p is non-ordinary, that is, ιp(ap( f )) is not a p-adic unit. Hence there is no unit rootpart in e f (MJ ⊗Zp L). Since we are interested in the f -part, we may assume that (1− e f )Nα is of the form(Munit

J ⊗L)⊕N′ with F-stable N′. Then Nα is F-stable and adequate by Corollary 2 and Proposition 6. Wecan also use the norm construction of the p-adic height pairing for Nα = Φ(Nα).

Let J be the formal group of the proper smooth model of J0(N) over Zp. Then by the Albanese map wecan embed

Γ (X0(N),Ω 1X0(N)/Q)⊗Qp = LJ⊗Qp ⊂MJ⊗Qp

and Fil1MJ⊗Qp =V LJ⊗Qp. We also have e f (MJ⊗Zp L) = e f (MJ⊗Zp L).Let Hn be the ring class field for the order Z+ pnOK . Let K be the completion of H = H0 at a

place w over p and let Kn be the completion of Hn at the place over w. (The place w is not necessarythe place compatible with ιp.) Since p splits in K , the extension Kn/Qp is abelian and K∞ = ∪nKn is aZp-extension over K1 obtained by a Lubin-Tate formal group over Zp of height 1. We denote the integerring of K by W and the residue field by k. By Proposition 5.1 of [15], there is an admissible norm system(cn)n ∈ e f (L⊗∏

∞n=1 J(mn)) obtained from Heegner points such that

2 Since the local height is defined on divisors and not on divisor classes, we have to give an appropriate meaning of “thef -part".


Trn+1/ncn+1−ap( f )cn + cn−1 = 0

for n ≥ 2 in J(mn)L with Tr1/0 c1 = zH .3 As in Example 1, this system is interpolated by an element ofZN(k[[T ]]) ⊂ J(k[[T ]])L with a W ⊗Zp L-submodule N ⊂ MJ/W ⊗ L such that e f N = e f (LJ ⊗W ⊗ L).Now we put

cn,α := Trn+1/ncn+1− pcn⊗α−1 ∈ J(mn)L.

Then by Proposition 8, the system (cn,α) is an admissible norm system satisfying

Trn+1/ncn+1,α = αcn,α

for n ≥ 1 and interpolated by an element of ZNα⊗Qp K(k[[T ]]). Note that e f Nα ⊗Qp K is given as (V −α)e f (Fil1MJ⊗W⊗L). By Lemma 2, we can take an admissible lift (cn,α)∈ L⊗∏

∞n=1 X

(n−1)D (mn) satisfying

Trn+1/ncn+1,α = α cn,α . Then we have

snormD,w,N(zH ) = Tr1/0 c1,α = α

−n+1Trn/0 cn,α

for any n with n≡ 1 mod ν . Then the proof in [15] works in our setting.

Acknowledgements The author thanks Kazuto Ota for discussion. This paper was written when the author was visitingl’Institute de Mathématiques de Jussieu in 2013, supported by Strategic Young Researcher Overseas Visits Program for Ac-celerating Brain Circulation by JSPS. He would like to thank sincerely Jan Nekovár for his hospitality. He is grateful to thereferee for carefully reading the paper. The research is also partly supported by Inamori Foundation and Grant-in-Aid forYoung Scientists (A) 25707001 by JSPS.

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Iwasawa modules arising from deformation spaces ofp-divisible formal group laws

Jan Kohlhaase

Abstract Let G be a p-divisible formal group law over an algebraically closed field of characteristic p.We show that certain equivariant vector bundles on the universal deformation space of G give rise to pseu-docompact modules over the Iwasawa algebra of the automorphism group of G. Passing to global rigidanalytic sections, we obtain representations which are topologically dual to locally analytic representations.In studying these, one is led to the consideration of divided power completions of universal envelopingalgebras. The latter seem to constitute a novel tool in p-adic representation theory.

0 Introduction

Let p be a prime number, and let k be an algebraically closed field of characteristic p. Let W =W (k) denotethe ring of Witt vectors with coefficients in k, and let K denote the quotient field of W . We fix a p-divisiblecommutative formal group law G of height h over k and denote by R := Rdef

G the universal deformation ringof G representing isomorphism classes of deformations of G to complete noetherian local W -algebras withresidue class field k. Denote by G the universal deformation of G to R and by Lie(G) the Lie algebra of G.For any integer m, the m-th tensor power Lie(G)⊗m of Lie(G) can be viewed as the space of global sectionsof a vector bundle on the universal deformation space Spf(R) which is equivariant for a natural action of theautomorphism group Γ := Aut(G) of G.

If G is of dimension one, then the formal scheme Spf(R) is known as the moduli space of Lubin-Tate. Itplays a crucial role in Harris’ and Taylor’s construction of the local Langlands correspondence for GLh(Qp).Moreover, the Γ -representations Lie(G)⊗m and their cohomology figure prominently in stable homotopytheory. Still assuming G to be one dimensional, a detailed study of the Γ -representation R was given in[13]. For h = 2 it led to the computation of the continuous Γ -cohomology of R, relying on the foundationalwork of Devinatz, Gross, Hopkins and Yu. The only prior analysis of p-adic representations stemming fromequivariant vector bundles on deformation spaces of p-divisible formal groups concern the p-adic symmet-ric spaces of Drinfeld. These were studied extensively by Morita, Orlik, Schneider and Teitelbaum (cf. [18],[25] and our remarks at the end of section 2).

The aim of the present article is to generalize and strengthen some of the results of Gross and Hopkins in[9] and of the author in [13]. To this end, section 1 and the first part of section 2 give a survey of the theoryof p-divisible commutative formal group laws. This includes the classification results of Dieudonné, Lazardand Manin, as well as the deformation theoretic results of Cartier, Lubin, Tate and Umemura. It followsfrom the work of Dieudonné and Manin that the group Γ is a compact Lie group over Qp (cf. Corollary 1).

In the second part of section 2, we prove that the action of Γ on Lie(G)⊗m extends to the Iwasawa al-gebra Λ :=W JΓ K of Γ over W . This gives Lie(G)⊗m the structure of a pseudocompact module over Λ (cf.

Jan KohlhaaseMathematisches Institut, Universität Münster e-mail: [email protected]

209

[email protected]

210 Jan Kohlhaase

Corollary 2 and Theorem 5). In section 3, we pass to the global rigid analytic sections (Lie(G)⊗m)rig of ourvector bundles and show that the action of Γ extends to a continuous action of the locally analytic distribu-tion algebra D(Γ ) of Γ over K. As a consequence, the action of Γ on the strong continuous K-linear dualof (Lie(G)⊗m)rig is locally analytic in the sense of Schneider and Teitelbaum (cf. Theorem 6 and Theorem7).

We note that the continuity and the differentiability of the action of Γ on Rrig were first proven by Grossand Hopkins if G is of dimension one (cf. [9], Proposition 19.2 and Proposition 24.2). Using the structuretheory of the algebra D(Γ ), we arrive at a more precise result for arbitrary m and G, avoiding the use of theperiod morphism. Our approach essentially relies on a basic lifting lemma for endomorphisms of G whichis also at the heart of the strategy followed by Gross and Hopkins (cf. Lemma 1 and Proposition 1).

A major question that we have to leave open concerns the coadmissibility of the D(Γ )-modules(Lie(G)⊗m)rig in the sense of [24], section 6. Taking sections over suitable affinoid subdomains of Spf(R)rig,it is related to the finiteness properties of the resulting Banach spaces as modules over certain Banach com-pletions of Λ ⊗W K. In section 4, we assume G to be of dimension one and consider the restriction of(Lie(G)⊗m)rig to an affinoid subdomain of Spf(R)rig over which the period morphism of Gross and Hopkinsis an open immersion. By spelling out the action of the Lie algebra of Γ , we show that one naturally obtainsa continuous module over a complete divided power enveloping algebra Udp

K (g) constructed by Kostant (cf.Theorem 8). Here g is a Chevalley order in the split form of the Lie algebra of Γ . If h = 2 and m≥−1 thenin fact (Lie(G)⊗m)rig gives rise to a cyclic module over Udp

K (g) (cf. Theorem 9). This result might indicatethat (Lie(G)⊗m)rig does not give rise to a coherent sheaf for the Fréchet-Stein structure of D(Γ ) consideredin [24], section 5 (cf. Remark 3).

In [9], Gross and Hopkins consider formal modules of dimension one and finite height over the valuationring o of an arbitrary non-archimedean local field. The case of p-divisible formal groups corresponds to thecase o = Zp. However, neither the deformation theory nor the theory of the period morphism have beenworked out in detail for formal o-modules of dimension strictly greater than one. This is why we restrict toone dimensional formal groups in section 4 and to p-divisible formal groups throughout.

Conventions and notation. If S is a commutative unital ring, if r is a positive integer, and if X =(X1, . . . ,Xr) is a family of indeterminates, then we denote by SJXK = SJX1, . . . ,XrK the ring of for-mal power series in the variables X1, . . . ,Xr over S. We write f = f (X) = f (X1, . . . ,Xr) for an elementf ∈ SJXK. If n = (n1, . . . ,nr) ∈ Nr is an r-tuple of non-negative integers then we set |n| := n1 + . . .+nr andXn := Xn1

1 · · ·Xnrr . If i and j are elements of a set then we denote by δi j the Kronecker symbol with value

1 ∈ S if i = j and 0 ∈ S if i 6= j. If h is a Lie algebra over S then we denote by U(h) the universal envelopingalgebra of h over S. Throughout the article, p will denote a fixed prime number.

1 Formal group laws

Let R be a commutative unital ring, and let d be a positive integer. A d-dimensional commutative formalgroup law (subsequently abbreviated to formal group) is a d-tuple G = (G1, . . . ,Gd) of formal power seriesin 2d variables Gi ∈ RJX ,Y K = RJX1, . . . ,Xd ,Y1, . . . ,YdK, satisfying

(F1) Gi(X ,0) = Xi,(F2) Gi(X ,Y ) = Gi(Y,X), and(F3) Gi(G(X ,Y ),Z) = Gi(X ,G(Y,Z))

for all 1 ≤ i ≤ d. It follows from the formal implicit function theorem (cf. [11], A.4.7) that for a given d-dimensional commutative formal group G there exists a unique d-tuple ιG ∈ RJXKd of formal power serieswith trivial constant terms such that

Gi(X , ιG(X)) = 0 for all 1≤ i≤ d

Iwasawa modules arising from deformation spaces of p-divisible formal group laws 211

(cf. also [28], Korollar 1.5). Thus, if S is a commutative R-algebra, and if I is an ideal of S such that S isI-adically complete, then the set Id becomes a commutative group with unit element (0, . . . ,0) via

x+G y := G(x,y) and − x := ιG(x).

Example 1. Let R=Z and d = 1. The formal group Ga(X ,Y )=X+Y is called the one dimensional additiveformal group. We have ιGa

(X) = −X . The formal group Gm(X ,Y ) = (1+X)(1+Y )− 1 is called the onedimensional multiplicative formal group. We have ιGm

(X) = ∑∞n=1(−X)n.

Let G and H be formal groups over R of dimensions d and e, respectively. A homomorphism from G to His an e-tuple ϕ = (ϕ1, . . . ,ϕe) of power series ϕi ∈ RJXK = RJX1, . . . ,XdK in d-variables over R with trivialconstant terms, satisfying

ϕ(G(X ,Y )) = H(ϕ(X),ϕ(Y )).

If ϕ : G → G′ and ψ : G′ → G′′ are homomorphisms of formal groups then we define ψ ϕ through(ψ ϕ)(X) := ψ(ϕ(X)). This is a homomorphism from G to G′′. We let End(G) denote the set of en-domorphisms of a d-dimensional commutative formal group G over R, i.e. of homomorphisms from Gto G. It is a ring with unit 1G = X = (X1, . . . ,Xd), in which addition and multiplication are defined by(ϕ +G ψ)(X) := G(ϕ(X),ψ(X)), (−ϕ)(X) := ιG(ϕ(X)) and ψ ·ϕ := ψ ϕ . In particular, End(G) is a Z-module. Given m ∈ Z, we denote by [m]G ∈ RJXKd the corresponding endomorphism of G. We denote byAut(G) the automorphism group of G, i.e. the group of units of the ring End(G).

Denoting by (X) the ideal of RJXK generated by X1, . . . ,Xd , the free R-module

Lie(G) := HomR((X)/(X)2,R)

of rank d = dim(G) is called the Lie algebra of G (or its tangent space at 1G). It is an R-Lie algebra for thetrivial Lie bracket. Non-commutative Lie algebras occur only for non-commutative formal groups (cf. [28],Kapitel I.7). An R-basis of Lie(G) is given by the linear forms ( ∂

∂Xi)1≤i≤d sending f +(X)2 to ∂ f

∂Xi(0). Here

∂ f∂Xi

denotes the formal derivative of the power series f with respect to the variable Xi.

Any homomorphism ϕ : G→H of formal groups as above gives rise to an R-linear ring homomorphismϕ∗ : RJY1, . . . ,YeK→ RJX1, . . . ,XdK, determined by ϕ∗(Yi) = ϕi for all 1≤ i≤ e. It is called the comorphismof ϕ . It maps (Y ) to (X), hence (Y )2 to (X)2, and therefore induces an R-linear map

Lie(ϕ) : Lie(G)−→ Lie(H)

via Lie(ϕ)(δ )(h+(Y )2) := δ (ϕ∗(h)+(X)2). In the R-bases ( ∂

∂Xi)i (resp. ( ∂

∂Y j) j) of Lie(G) (resp. Lie(H)),

the map Lie(ϕ) is given by the Jacobian matrix ( ∂ϕi∂X j

(0))i, j ∈ Re×d of ϕ . If ϕ : G→G′ and ψ : G′→G′′ arehomomorphisms of formal groups, then (ψ ϕ)∗ = ϕ∗ ψ∗ and Lie(ψ ϕ) = Lie(ψ) Lie(ϕ). If H = Gthen one can use (F1) to show that the map (ϕ 7→ Lie(ϕ)) : End(G)→ EndR(Lie(G)) is a homomorphismof rings. In particular, Lie(G) becomes a module over End(G) and we have Lie([m]G) = m · idLie(G) for anyinteger m.

If p is a prime number and if R is a complete noetherian local ring of residue characteristic p, then ahomomorphism ϕ : G→H of formal groups is called an isogeny if the comorphism ϕ∗ makes RJXK a finitefree module over RJY K (cf. [26], section 2.2). Of course, this can only happen if d = e. A formal group Gover a complete noetherian local ring R with residue characteristic p is called p-divisible, if the homomor-phism [p]G : G→ G is an isogeny. In this case the rank of RJXK over itself via [p]∗G is a power of p, say ph

(cf. [26], section 2.2; this result can also be deduced from [28], Satz 5.3). The integer h =: ht(G) is calledthe height of the p-divisible formal group G.

If R = k is a perfect field of characteristic p, the necessary tools to effectively study the category ofp-divisible commutative formal groups over k were first developed by Dieudonné (cf. [6], Chapter III). Hismethods were later generalized by Cartier in order to describe commutative formal groups over arbitraryrings (cf. [16], Chapters III & IV, or [28], Chapters III & IV).

212 Jan Kohlhaase

Sticking to the case of a perfect field k of characteristic p, we denote by W := W (k) the ring of Wittvectors over k. Let σ = (x 7→ xp) denote the Frobenius automorphism of k, as well as its unique lift to a ringautomorphism of W . Recall that a σ−1-crystal over k is a pair (M,V ), consisting of a finitely generated freeW -module M and a map V : M→M which is σ−1-linear, i.e. which is additive and satisfies

V (am) = σ−1(a)V (m) for all a ∈W, m ∈M.

We shall be interested in those σ−1-crystals (M,V ) which satisfy the following two extra conditions (hereD stands for Dieudonné):

(D1) pM ⊆V (M)(D2) V mod p is a nilpotent endomorphism of M/pM.

For the following fundamental result cf. [28], page 109.

Theorem 1 (Dieudonné). If k is a perfect field of characteristic p then the category of p-divisible commu-tative formal groups over k is equivalent to the category of σ−1-crystals over k, satisfying (D1) and (D2).

Let W [F,V ] be the non-commutative ring generated by two elements F and V over W subject to therelations

V F = FV = p, Va = σ−1(a)V and Fa = σ(a)F for all a ∈W.

The equivalence of Theorem 1 associates with a p-divisible commutative formal group G its (covariant)Cartier-Dieudonné module MG. This is a V -adically separated and complete module over W [V,F ] such thatthe action of V is injective. Since G is p-divisible, also the action of F is injective, and the underlying W -module of MG is finitely generated and free. In particular, the pair (MG,V ) is a σ−1-crystal over k, satisfyingpMG = V FMG ⊆ V MG, i.e. condition (D1). Condition (D2) follows from the V -adic completeness of MG.We also note that V and F give rise to a short exact sequence

0 // MG/FMGV // MG/pMG // MG/V MG // 0,

of k-vector spaces in which dimk(MG/pMG) = ht(G) and dimk(MG/V MG) = dim(G).

Conversely, if (M,V ) is a σ−1-crystal over k satisfying (D1), then V is injective. In fact, (D1) implies thatV becomes surjective (and hence bijective) over the quotient field K of W . Setting F :=V−1 p, the W -moduleM becomes a module over W [F,V ] which is V -adically separated and complete if condition (D2) is satisfied.

Recall that a σ−1-isocrystal over k is a pair (N, f ) consisting of a finite dimensional K-vector spaceN and a σ−1-linear bijection f : N → N. If (M,V ) is a σ−1-crystal over k which satisfies (D1) then(M⊗W K,V ⊗ idK) is a σ−1-isocrystal over k. The σ−1-isocrystal which in this way is associated with theCartier-Dieudonné module of a p-disivible commutative formal group G over k, classifies G up to isogeny(cf. [28], Satz 5.26 and the remarks on page 110; alternatively, consult [6], Chapter IV.1).

Given integers r and s with r > 0, consider the σ−1-isocrystal over k given by (K[t]/(tr− ps), t σ). HereK[t] denotes the usual commutative polynomial ring in the variable t over K on which σ acts coefficientwise.If k is algebraically closed, we have the following fundamental classification result of Dieudonné and Manin(cf. [28], Satz 6.29, [6], Chapter IV.4, and [16], Proposition VI.7.42).

Theorem 2 (Dieudonné-Manin). If k is an algebraically closed field of characteristic p then the categoryof σ−1-isocrystals over k is semisimple. The simple objects are given by the σ−1-isocrystals (K[t]/(tr −ps), t σ), where r and s are relatively prime integers with r > 0.

To a pair (r,s) of integers as in Theorem 2 corresponds a particular p-divisible commutative formal groupGrs over k inside the isogeny class determined by the σ−1-isocrystal (K[t]/(tr− ps), t σ). According to[16], Proposition VI.7.42, the endomorphism ring of Grs is isomorphic to the maximal order of the centraldivision algebra of invariant s

r +Z ∈Q/Z and dimension r2 over Qp.


Corollary 1. If G is a p-divisible commutative formal group over an algebraically closed field k of char-acteristic p then the endomorphism ring End(G) of G is an order in a finite dimensional semisimple Qp-algebra. Endowing End(G) with the p-adic topology and the automorphism group Aut(G) of G with theinduced topology, Aut(G) is a compact Lie group over Qp.

Proof. That End(G) is a p-adically separated and torsion free Zp-module can easily be proved directly,using that G is p-divisible. It also follows from the fact that the Cartier-Dieudonné module of G is free overW . According to Theorem 2 and the subsequent remarks there are central division algebras D1, . . . ,Dn overQp and natural numbers m1, . . . ,mn such that

End(G)⊗Zp Qp 'Mat(m1×m1,D1)× . . .×Mat(mn×mn,Dn)

as Qp-algebras. Since End(G) is p-adically separated, it is bounded in End(G)⊗Zp Qp. Thus, it is a lattice ina finite dimensional Qp-vector space and must be finitely generated over Zp. This proves the first assertion.Endowing End(G) with the p-adic topology, it becomes a topological Zp-algebra and Aut(G) becomes acompact topological group for the subspace topology. By the above arguments, it is isomorphic to an opensubgroup of ∏

ni=1 GLmi(Di), hence naturally carries the structure of a Lie group over Qp.

2 Deformation problems and Iwasawa modules

We continue to denote by k a fixed algebraically closed field of characteristic p. We also fix a p-divisiblecommutative formal group G of dimension d over k. Denote by W =W (k) the ring of Witt vectors of k andby Ck the category of complete noetherian commutative local W -algebras with residue class field k. Let Rbe an object of Ck and let m be the maximal ideal of R. A deformation of G to R is a pair (G′,ρG′), whereG′ is commutative formal group over R and ρG′ : G→ G′modm is an isomorphism of formal groups overk. Two deformations (G′,ρG′) and (G′′,ρG′′) of G to R are said to be isomorphic if there is an isomorphismf : G′→ G′′ of formal groups over R such that the diagram

G′modm

f modm

G

ρG′::

ρG′′ $$G′′modm

is commutative. Let DefG denote the functor from Ck to the category Sets of sets which associates with anobject R of Ck the set of isomorphism classes of deformations of G to R. If dim(G) = 1, then the followingtheorem was first proved by Lubin and Tate (cf. [17], Theorem 3.1), building on the work of Lazard. It waslater generalized by Cartier und Umemura, independently (cf. [5] and [27]).

Theorem 3. The functor DefG : Ck → Sets is representable, i.e. there is an object RdefG of Ck and a defor-

mation G of G to RdefG with the following universal property. For any object R of Ck and any deformation

(G′,ρG′) of G to R there is a unique W-linear local ring homomorphism ϕ : RdefG → R and a unique iso-

morphism [ϕ] : ϕ∗(G,ρG)' (G′,ρG′) of deformations of G to R.1 If h = ht(G) and d = dim(G) denote theheight and the dimension of G, respectively, then the W-algebra Rdef

G is non-canonically isomorphic to thepower series ring W Ju1, . . . ,u(h−d)dK in (h−d)d variables over W.

It follows from the universal property of the deformation (G,ρG) that the automorphism group Aut(G)of G acts on the universal deformation ring Rdef

G by W -linear local ring automorphisms. Indeed, given γ ∈Aut(G), there is a unique W -linear local ring endomorphism γ of Rdef

G and a unique isomorphism [γ] :γ∗(G,ρG)' (G,ρG γ) of deformations of G to Rdef

G . It follows from the uniqueness that the resulting mapAut(G)→ End(Rdef

G ) factors through a homomorphism

1 Here ϕ∗(G,ρG) = (ϕ∗(G),ρG), where ϕ∗G is obtained by applying ϕ to the coefficients of G. Since ϕ induces an isomor-phism between the residue class fields of Rdef

G and R, we may identify GmodmRdefG

and ϕ∗Gmodm.

214 Jan Kohlhaase

Aut(G)−→ Aut(RdefG )

of groups. It is this type of representation that we are concerned with in this article. To ease notation weshall denote by

R := RdefG

the universal deformation ring of our fixed p-divisible commutative formal group G over k. Let m denotethe maximal ideal of R. For any non-negative integer n we denote by Gn := Gmodmn+1 the reduction ofthe universal deformation G modulo the ideal mn+1 of R. We have G'G0 via ρG.

Lemma 1. If n is a non-negative integer then the ring homomorphism End(Gn+1)→ End(Gn), induced byreduction modulo mn+1, is injective.

Proof. The formal group Gn+1 is p-divisible because the comorphism [p]∗Gn+1is finite and free. Indeed, it is

so after reduction modulo m, and one can use [3], III.2.1 Proposition 14 and III.5.3 Théorème 1, to conclude.Since the ideal mn+1(R/mn+2) of R/mn+2 is nilpotent, the claim follows from the rigidity theorem in [28],Satz 5.30.

The preceding lemma allows us to regard all endomorphism rings End(Gn) as subrings of End(G0). Themain technical result of this section is the following assertion.

Proposition 1. For any non-negative integer n the subring End(Gn) of End(G0) contains pn End(G0).

Proof. We proceed by induction on n, the case n = 0 being trivial. Let n ≥ 1 and assume the assertion tobe true for n−1. Set Rn := R/mn+1. Let ϕ ∈ pn−1 End(G0)⊆ End(Gn−1) and choose a family ϕ ∈ RnJXKd

of power series with trivial constant terms such that ϕ modmnRn = ϕ . The d-tuple of power series [p]Gn ϕ

is then a lift of pϕ . We claim that it is an endomorphism of Gn.

Note first that [p]Gn ϕ depends only on ϕ and not on the choice of a lift ϕ . Indeed, if ϕ ′ is a second lift ofϕ with trivial constant terms, set ψ := ϕ ′− ϕ . Setting χ := (ϕ +ψ)−Gn ϕ , we have ϕ ′ = ϕ +Gn χ . Further,the power series χ satisfies χ modmn = ϕ−Gn−1 ϕ = 0, hence has coefficients in mnRn. Since pmn ⊆mn+1

and (mn)m ⊆mn+1 for any integer m≥ 2, we have [p]Gn χ = 0 and hence

[p]Gn ϕ′ = [p]Gn(ϕ +Gn χ) = ([p]Gn ϕ)+Gn ([p]Gn χ) = [p]Gn ϕ,

as desired.

If η ∈ RnJXKd is a family of power series with trivial constant terms, set δη := δη(X ,Y ) := η(X +Gn

Y )−Gn η(X)−Gn η(Y ). Since ϕ reduces to an endomorphism of Gn−1, the power series δϕ has coefficientsin mn. As above, this implies [p]Gn δϕ = 0 and thus

δ[p]Gnϕ = ([p]Gn ϕ)(X +Gn Y )−Gn ([p]Gn ϕ)(X)−Gn ([p]Gn ϕ)(Y )

= [p]Gn(δϕ) = 0.

As a consequence, [p]Gn ϕ ∈ End(Gn), and thus pϕ ∈ End(Gn). Since ϕ was arbitrary, we obtain thedesired inclusion pn End(G0)⊆ End(Gn).

According to Corollary 1, the group Aut(G) is a profinite topological group. A basis of open neighbor-hoods of its identity is given by the subgroups 1+ pn End(G) with n ≥ 1. If m denotes the maximal idealof the local ring R, the W -algebra R is a topological ring for the m-adic topology. We are now ready toprove the following result, a particular case of which was treated in [13], Proposition 3.1. The argument isborrowed from the proof of [9], Lemma 19.3. Let us put

Γ := Γ0 := Aut(G) and Γn := 1+ pn End(G) for n≥ 1.

Theorem 4. The action of Γ on R=RdefG is continuous in the sense that the map ((γ, f ) 7→ γ( f )) : Γ ×R→R

is a continuous map of topological spaces. Here Γ ×R carries the product topology. If n is a non-negativeinteger then the induced action of Γn on R/mn+1 is trivial.


Proof. As in the proof of [13], Proposition 3.1, it suffices to prove the second statement. Let γ ∈ Γn andconsider the deformation (Gn,ρG γ) of G to Rn = R/mn+1. Denote by prn : R→ Rn the natural projec-tion and let γn denote the unique ring homomorphism γn : R→ Rn for which there exists an isomorphismof deformations [γn] : (γn)∗(G,ρG) ' (Gn,ρG γ) (cf. Theorem 3). Note that also the ring homomorphismprn γ : R→ Rn admits an isomorphism of deformations (prn γ)∗(G,ρG) ' (Gn,ρG γ), namely the re-duction of [γ] modulo mn+1. By uniqueness, we must have γn = prn γ and [γn] = [γ] mod mn+1.

Since the map (σ 7→ ρG σ ρ−1G ) is a ring isomorphism End(G)→ End(G0), Proposition 1 shows that

ρG γ ρ−1G ∈Aut(Gn) and therefore defines an isomorphism of deformations (prn)∗(G,ρG) = (Gn,ρG)'

(Gn,ρG γ). By uniqueness again, we must have γn = prn γ = prn. This implies that γ acts trivially on Rnand that [γ] mod mn+1 = ρG γ ρ

−1G .

If H is a profinite topological group then we denote by

Λ(H) :=W JHK := lim←−n≥1,NEoH

(W/pnW )[H/N]

the Iwasawa algebra (or completed group ring) of H over W . The above projective limit runs over allpositive integers n and over all open normal subgroups N of H. If n and n′ are positive integers withn′ ≤ n, and if N and N′ are two open normal subgroups of H with N ⊆ N′, then the transition map(W/pnW )[H/N]→ (W/pn′W )[H/N′] is the natural homomorphism of group rings induced by the surjec-tive homomorphism H/N→ H/N′ of groups and the surjective ring homomorphism W/pnW →W/pn′W .Endowing each ring (W/pnW )[H/N] with the discrete topology, Λ(H) is a topological ring for the pro-jective limit topology. It is a pseudocompact ring in the terminology of [4], page 442, because each of therings (W/pnW )[H/N] is Artinian. Recall that a complete Hausdorff topological Λ(H)-module M is calledpseudocompact, if it admits a basis (Mi)i∈I of open neighborhoods of zero such that each Mi is a Λ(H)-submodule of M for which the Λ(H)-module M/Mi has finite length. For brevity, we will set

Λ := Λ(Aut(G)).

Corollary 2. The action of Aut(G) on R = RdefG extends to an action of Λ and gives R the structure of a

pseudocompact Λ -module.

Proof. Since R is m-adically separated and complete, we may consider the natural isomorphism

R' lim←−n≥0

R/mn+1.

According to Theorem 4, the action of the group ring W [Aut(G)] on R/mn+1 factors through (W/pn+1W )[Aut(G)/(1+pn End(G))] where 1+ pn End(G) is an open normal subgroup of Aut(G). Thus, R/mn+1 can be viewed as aΛ -module via the natural ring homomorphism Λ → (W/pn+1W )[Aut(G)/(1+ pn End(G))]. The transitionmaps in the above projective limit are then Λ -equivariant. This proves the first assertion.

As for the second assertion, the ideals mn+1 of R are open and Λ -stable, being the kernels of the Λ -equivariant projections R→ R/mn+1. They form a basis of open neighborhoods of zero of R, and the quo-tients R/mn+1 are even of finite length over W ⊆Λ .

Let Lie(G) denote the Lie algebra of the universal deformation G of G. This is a free module of rankd = dim(G) over R. Given γ ∈ Aut(G), we extend the ring automorphism γ : R→ R to an automorphismγ : RJXK→ RJXK by setting γ(Xi) = Xi for all 1≤ i≤ d. It induces a homomorphism γ : Lie(G)→ Lie(γ∗G)of additive groups. We define γ : Lie(G)→ Lie(G) as the composite of the two additive maps

Lie(G)γ // Lie(γ∗G)

Lie([γ]) // Lie(G),

with [γ] : γ∗G→G as above. Given a second element γ ′ ∈Aut(G), we define γ ′ : Lie(γ∗G)→ Lie(γ ′∗(γ∗G))as before. Further, γ ′∗[γ] : γ ′∗(γ∗G)→ γ ′∗G denotes the homomorphism obtained by applying γ ′ ∈ Aut(R) tothe coefficients of [γ] ∈ RJXKd . One readily checks that the diagram

216 Jan Kohlhaase

Lie(γ∗G)Lie([γ]) //

γ ′

Lie(G)

γ ′

Lie(γ ′∗(γ∗G))

Lie(γ ′∗[γ])// Lie(γ ′∗G)

is commutative. Further, the uniqueness assertion in Theorem 3 implies that [γ ′γ] = [γ ′] γ ′∗[γ]. Therefore,

(γ ′γ)∼ = Lie([γ ′γ]) (γ ′γ) = Lie([γ ′])Lie(γ ′∗[γ]) γ′ γ

= Lie([γ ′]) (γ ′ Lie([γ]) (γ ′)−1) γ′ γ = γ

′ γ.

As a consequence, we obtain an action of Aut(G) on the additive group Lie(G) which is semilinear for theaction on R in the sense that

γ( f ·δ ) = γ( f ) · γ(δ ) for all f ∈ R,δ ∈ Lie(G).

To ease notation, we will again write γ(δ ) for γ(δ ).

Given a positive integer m we denote by Lie(G)⊗m the m-fold tensor product of Lie(G) over R withitself. This is a free R-module of rank dm with a semilinear action of Aut(G) defined by

γ(δ1⊗·· ·⊗δm) := γ(δ1)⊗·· ·⊗ γ(δm).

We also set Lie(G)⊗0 := R and Lie(G)⊗m := HomR(Lie(G)⊗(−m),R) if m is a negative integer. In the lattercase Lie(G)⊗m is a free R-module of rank d−m with a semilinear action of Aut(G) defined through

γ(ϕ)(δ1⊗·· ·⊗δ−m) := γ(ϕ(γ−1(δ1)⊗·· ·⊗ γ−1(δ−m))).

For any integer m we endow the R-module Lie(G)⊗m with the m-adic topology for which it is Hausdorff andcomplete. By the semilinearity of the Aut(G)-action, the R-submodules mn Lie(G)⊗m are Aut(G)-stable forany non-negative integer n.

As an easy consequence of Proposition 1 and Theorem 4, we obtain the following result.

Theorem 5. Let m and n be integers with n ≥ 0. The action of Aut(G) on Lie(G)⊗m is continuous inthe sense that the structure map Aut(G)×Lie(G)⊗m → Lie(G)⊗m of the action is continuous. Here theleft hand side carries the product topology. The induced action of 1 + p2n+1 End(G) on the quotientLie(G)⊗m/mn+1 Lie(G)⊗m is trivial. In particular, the action of Aut(G) on Lie(G)⊗m extends to an ac-tion of Λ and gives Lie(G)⊗m the structure of a pseudocompact Λ -module.

Proof. As in the proof of Theorem 4 and Corollary 2, it suffices to show that the action of 1+ p2n+1 End(G)on Lie(G)⊗m/mn+1 Lie(G)⊗m is trivial. By definition of the action and Theorem 4 we may assume m = 1.Setting Gn =G mod mn+1, as before, we have Lie(G)/mn+1 Lie(G)=Lie(Gn). Since 2n+1≥ n, Theorem4 and its proof show that the map γ mod mn+1 : Lie(Gn)→ Lie(Gn) is given by Lie(ρG γ ρ

−1G ) where

ρG γ ρ−1G is contained in 1+ p2n+1 End(G0)⊆ 1+ pn+1 End(Gn) (cf. Proposition 1). Therefore, it suffices

to show that the natural action of 1+ pn+1 End(Gn) ⊂ End(Gn) on Lie(Gn) is trivial. However, if ϕ ∈End(Gn) and if δ ∈ Lie(Gn), then

Lie(1+ pn+1ϕ)(δ ) = δ + pn+1 Lie(ϕ)(δ ) = δ ,

because pn+1 ∈mn+1.

Before we continue, let us point out an important variant of the deformation problem considered above.It concerns the moduli problems considered by Rapoport and Zink (cf. [19]).

Let G be a fixed p-divisible group over the algebraically closed field k of characteristic p, i.e. an fppf -group scheme over Spec(k) for which multiplication by p is an epimorphism. Denoting by Nilp the category


of W -schemes on which p is locally nilpotent, let MG : Nilp→ Sets denote the set valued functor which as-sociates to an object S of Nilp the set of isomorphism classes of pairs (G′,ρG′), where G′ is a p-divisiblegroup over S and ρG′ : GS → G′

Sis a quasi-isogeny (cf. [19], Definition 2.8). Here S denotes the closed

subscheme of S defined by the sheaf of ideals pOS. According to [19], Theorem 2.16, the functor MG isrepresented by a formal scheme which is locally formally of finite type over Spf(W ). If G is a p-divisibleone dimensional commutative formal group law as in section 1, then MG is the disjoint union of open sub-schemes Mn

G, n ∈ Z, which are non-canonically isomorphic to Spf(RdefG ) (cf. [19], Proposition 3.79). The

reason is that any quasi-isogeny of height zero between one dimensional p-divisible formal group laws overk is an isomorphism.

One can generalize the moduli problem even further by considering deformations of p-divisible groupswith additional structures such as polarizations or actions by maximal orders in finite dimensional semisim-ple Qp-algebras (cf. [19], Definition 3.21). The corresponding deformation functors are again representable,as was proven by Rapoport and Zink (cf. [19], Theorem 3.25). An important example was studied by Drin-feld (cf. [19], 3.58). The generic fiber of the representing formal scheme is known as Drinfeld’s upper halfspace over K. Instead of continuous representations of Aut(G) as in Theorem 4, it gives rise to an importantclass of p-adic locally analytic representations in the sense of Schneider and Teitelbaum. This particularclass of representations was studied extensively by Morita, Orlik, Schneider and Teitelbaum (cf. [18] and[25]). It found aritheoremetic applications to the de Rham cohomology of varieties which are p-adically uni-formized by Drinfeld’s upper half space (cf. [KS95]). In the next section we shall see that the deformationspaces we consider here give rise to locally analytic representations, as well.

3 Rigidification and local analyticity

We keep the notation of the previous section and denote by k an algebraically closed field of characteristicp and by G a fixed commutative p-divisible formal group over k. Let h and d denote the height and thedimension of G, respectively. We denote by W the ring of Witt vectors of k and by K the quotient field ofW . We let R = Rdef

G denote the universal deformation ring of G (cf. Theorem 3).

According to Theorem 3, the rigidification Spf(R)rig of the formal scheme Spf(R) in the sense of Berth-elot (cf. [10], section 7) is isomorphic to the (h−d)d-dimensional rigid analytic open unit polydisc B(h−d)d

Kover K. We let

Rrig := O(Spf(R)rig)

denote the ring of global rigid analytic functions on Spf(R)rig. Any isomorphism R 'W JuK of local W -algebras extends to an isomorphism

Rrig ' ∑α∈N(h−d)d

cα uα | cα ∈ K and lim|α|→∞

|cα |r|α| = 0 for all 0 < r < 1

of K-algebras, where | · | denotes the p-adic absolute value on K. This allows us to view Rrig as a topologicalK-Fréchet algebra whose topology is defined by the family of norms || · ||`, given by

||∑α

cα uα ||` := supα

|cα |p−|α|/`

for any positive integer `. Letting Rrig` denote the completion of Rrig with respect to the norm || · ||`, the

K-algebra Rrig` can be identified with the ring of rigid analytic functions on the affinoid subdomain

B(h−d)d` := x ∈ Spf(R)rig | |ui(x)| ≤ p−1/` for all 1≤ i≤ (h−d)d

of Spf(R)rig. Further, Rrig ' lim←−`Rrig` is the topological projective limit of the K-Banach algebras Rrig

` . Infact, by a cofinality argument and [1], 6.1.3 Theorem 1, Rrig is the topological projective limit of the system

218 Jan Kohlhaase

of affinoid K-algebras corresponding to any nested admissible open affinoid covering of Spf(R)rig.

By functoriality, the automorphism group Γ = Aut(G) of G acts on Spf(R)rig by automorphisms of rigidanalytic K-varieties. This gives rise to an action of Γ on Rrig by K-linear ring automorphisms. By the abovecofinality argument, any of these automorphisms is continuous. The goal of this section is to show that theinduced action on the strong topological K-linear dual of Rrig is locally analytic in the sense of Schneiderand Teitelbaum (cf. [23], page 451).

Fix an algebraic closure Kalg of K. According to [10], Lemma 7.19, the maximal ideals of the ringRK := R⊗W K are in bijection with the points of Spf(R)rig. It follows from [1], 7.1.1 Proposition 1, that thelatter are in bijection with the Gal(Kalg|K)-orbits of

B(h−d)dK (Kalg) := x ∈ (Kalg)(h−d)d | |xi|< 1 for all 1≤ i≤ (h−d)d.

A point x representing one of these orbits corresponds to the kernel of the surjective K-linear ring homo-morphism RK → K(x) := K(x1, . . . ,x(h−d)d)⊆ Kalg, sending f (u) to f (x).

The following result constitutes the technical heart of this section. It is a straightforward generalizationof [9], Lemma 19.3.

Proposition 2. Let n and ` be integers with n ≥ 0 and ` ≥ 1. If γ ∈ Γn and if f ∈ Rrig then ||γ( f )− f ||` ≤p−n/`|| f ||`.

Proof. First assume f = ui for some 1≤ i≤ (h−d)d. If

B(h−d)d` (Kalg) := x ∈ (Kalg)(h−d)d | |xi| ≤ p−1/` for all 1≤ i≤ (h−d)d,

then ||g||` = sup|g(x)| | x ∈ B(h−d)d` (Kalg) for any g ∈ Rrig. Thus, we need to see that if x ∈ B(h−d)d

` (Kalg)

and if y := x · γ = γ(u)(x), then |xi− yi| ≤ p−(n+1)/`.

Denoting by W alg the valuation ring of Kalg, consider the commutative diagram

Rγ //

!!

R

W alg

of homomorphisms of W -algebras, in which the left and right oblique arrow is given by evaluation at y andx, respectively. Choosing z∈W alg with |z|= p−1/`, we have x j ∈ zW alg for any j. Further, p∈ zW alg because`≥ 1. As a consequence, the right oblique arrow maps mR to zW alg. Note that γ(u j) ∈mR, so that we obtainy j = u j(x · γ) = γ(u j)(x) ∈ zW alg, as well. Therefore, also the left oblique arrow maps mR to zW alg. Nowconsider the induced diagram

R/mn+1R

γ //

&&

R/mn+1R

xxW alg/(zn+1).

According to Theorem 4, the upper horizontal arrow is the identity. It follows that xi− yi ∈ zn+1W alg, i.e.|xi− yi| ≤ p−(n+1)/`, as required. In particular, γ stabilizes B(h−d)d

` (Kalg) and therefore is an isometry forthe norm || · ||` on Rrig.

To prove the proposition, the continuity of γ allows us to assume f = uα for some α ∈ N(h−d)d . Theassertion is trivial for |α|= 0. If |α|> 0 choose an index i with αi > 0. Define β through β j := α j if j 6= iand βi := αi−1. If x ∈ B(h−d)d

` (Kalg) and if y = x · γ , then


|γ(uα)(x)−uα(x)| = |yα − xα |= |yiyβ − xixβ |≤ max|yi||yβ − xβ |, |yi− xi||xβ |.

Here |yi||yβ − xβ | ≤ p−1/`||γ(uβ )− uβ ||` ≤ p−(n+1)/`||uβ ||` = p−n/`||uα ||` by the induction hypothesis.Further, |yi−xi||xβ | ≤ p−(n+1)/`p−|β |/` = p−n/`||uα ||`, as seen above. Thus, we obtain |γ(uα)(x)−uα(x)| ≤p−n/`||uα ||` for all x ∈ B(h−d)d

` (Kalg). This proves the proposition.

A topological group is a Lie group over Qp if and only if it contains an open subgroup which is a uniformpro-p group (cf. [8], Definition 4.1 and Theorem 8.32). For the compact p-adic Lie group Γ = Aut(G) wehave the following more precise result. We let

ε := 1 if p > 2 and ε := 2 if p = 2.

Lemma 2. For any non-negative integer n we have Γpn

ε = Γε+n. The open subgroup Γε+n of Γ is a uniformpro-p group.

Proof. As for the first assertion, the proofs of [8], Lemma 5.1 and Theorem 5.2, can be copied word byword on replacing Md(Zp) by End(G) and GLd(Zp) by Aut(G). Further, Γε+n is a powerful pro-p group by[8], Theorem 3.6 (i) and the remark after Definition 3.1. That it is uniform follows from [8], Theorem 3.6(ii), and the first assertion.

Fix an integer n ≥ ε . By Lemma 2 and [8], Theorem 3.6, the group Γn/Γn+1 is a finite dimensional Fp-vector space. Choosing elements γ1, . . . ,γr ∈ Γn whose images modulo Γn+1 form an Fp-basis of Γn/Γn+1,[8], Theorem 4.9, shows that (γ1, . . . ,γr) is an ordered basis of Γn in the sense that the map Zr

p→Γn, sending

λ to γλ11 · · ·γλr

r , is a homeomorphism.

Set bi := γi− 1 ∈ Λ(Γn) and bα := bα11 · · ·bαr

r for any α ∈ Nr. By [8], Theorem 7.20, any element δ ∈Λ(Γn) admits a unique expansion of the form

λ = ∑α∈Nr

dα bα with dα ∈W for all α ∈ Nr.

For any `≥ 1 this allows us to define the K-norm || · ||` on the algebra Λ(Γn)K := Λ(Γn)⊗W K through

||∑α

dα bα ||` := supα

|dα |p−ε|α|/`.

Remark 1. A more accurate notation would be the symbol || · ||(n)` for the above norm on Λ(Γn)K . It does

generally not coincide with the restriction of || · ||(m)` to Λ(Γn)K ⊆ Λ(Γm)K if n ≥ m. However, there is

an explicit rescaling relation between the families of norms (|| · ||(n)` )` and (|| · ||(m)` )` on Λ(Γn)K (cf. [20],

Proposition 6.2). Since we will never work with two different groups Γn and Γm at once, we decided to easenotation and use the somewhat ambiguous symbol || · ||`.

By [20], Proposition 2.1 and [24], Proposition 4.2, the norm || · ||` on Λ(Γn)K is submultiplicative when-ever `≥ 1. As a consequence, the completion

Λ(Γn)K,` = ∑α

dα bα | dα ∈ K, lim|α|→∞

|dα |p−ε|α|/` = 0

of Λ(Γn)K with respect to || · ||` is a K-Banach algebra. The natural inclusions Λ(Γn)K,`+1→Λ(Γn)K,` endowthe projective limit

D(Γn) := lim←−Λ(Γn)K,`

with the structure of a K-Fréchet algebra. As is explained in [24], section 4, a theorem of Amice allows usto identify it with the algebra of K-valued locally analytic distributions on Γn. Similarly, we denote by D(Γ )the algebra of K-valued locally analytic distributions on Γ (cf. [23], section 2).

220 Jan Kohlhaase

Theorem 6. For any integer ` ≥ 1 the action of Γε on Rrig extends to Rrig` and makes Rrig

` a topologicalBanach module over the K-Banach algebra Λ(Γε)K,`. The action of Γ on Rrig extends to a jointly continuousaction of the K-Fréchet algebra D(Γ ). The action of Γ on the strong continuous K-linear dual (Rrig)′b ofRrig is locally analytic in the sense of [23], page 451.

Proof. First, we prove by induction on |α| that ||bα f ||` ≤ ||bα ||`|| f ||` for any f ∈ Rrig. This is clear if|α| = 0. Otherwise, let i be the minimal index with αi > 0 and define β through β j = α j if j 6= i andβi := αi−1. In this case, Proposition 2 and the induction hypothesis imply

||bα f ||` = ||(γi−1)bβ f ||` ≤ p−ε/`||bβ f ||`≤ p−ε/`p−ε|β |/`|| f ||` = ||bα ||`|| f ||`,

as required. This immediately gives ||λ · f ||` ≤ ||λ ||`|| f ||` for all λ ∈Λ(Γε)K and f ∈ RK . Thus, the multi-plication map Λ(Γε)K ×RK → RK is continuous, if Λ(Γε)K and RK are endowed with the respective || · ||`-topologies, and if the left hand side carries the product topology. Since RK is dense in Rrig

` , we obtain a mapΛ(Γε)K,`×Rrig

` → Rrig` by passing to completions. By continuity, it gives Rrig

` the structure of a topologicalBanach module over Λ(Γε)K,`.

Passing to the projective limit, we obtain a continuous map D(Γε)×Rrig→ Rrig, giving Rrig the structureof a jointly continuous module over D(Γε). Since D(Γ ) is topologically isomorphic to the locally convexdirect sum ⊕γΓε∈Γ /Γε

γD(Γε) (cf. [23], page 447 bottom), Rrig is a jointly continuous module over D(Γ ).

It follows from [21], Proposition 19.9 and the arguments proving the claim on page 98, that the K-Fréchetspace Rrig is nuclear. Therefore, [23], Corollary 3.4, implies that the locally convex K-vector space (Rrig)′bis of compact type and that the action of Γ obtained by dualizing is locally analytic.

Using Theorem 5, the preceding result can be generalized as follows. Fixing an integer m, the free R-module Lie(G)⊗m gives rise to a locally free coherent sheaf on Spf(R). For any positive integer ` we denoteby (Lie(G)⊗m)

rig` the sections of its rigidification over the affinoid subdomain B(h−d)d

` of Spf(R)rig. This isa free Rrig

` -module for which the natural Rrig` -linear map

Rrig` ⊗R Lie(G)⊗m −→ (Lie(G)⊗m)

rig`

is bijective (cf. [10], 7.1.11). We denote by (Lie(G)⊗m)rig the space of global sections of the rigidificationof Lie(G)⊗m over Spf(R)rig. This is a free Rrig-module for which the natural Rrig-linear maps

Rrig⊗R Lie(G)⊗m −→ (Lie(G)⊗m)rig −→ lim←−(Lie(G)⊗m)rig` (1)

are bijective. Further, (Lie(G)⊗m)rig' (Lie(G)rig)⊗m, where the latter tensor products and dualities are withrespect to Rrig.

By functoriality, the group Γ = Aut(G) acts on (Lie(G)⊗m)rig in such a way that the left map in (1) be-comes Γ -equivariant for the diagonal action on the left. In particular, it is semilinear for the action of Γ onRrig. We endow (Lie(G)⊗m)rig and (Lie(G)⊗m)

rig` with the natural topologies of finitely generated modules

over Rrig and Rrig` , respectively. This makes them a nuclear K-Fréchet space and a K-Banach space, respec-

tively. The right map in (1) is then a topological isomorphism for the projective limit topology on the right.With the same cofinality argument as for Rrig one can show that any element of Γ acts on (Lie(G)⊗m)rig

through a continuous K-linear automorphism.

Theorem 7. Let m be an integer. For any integer ` ≥ 1 the action of Γ2ε−1 on (Lie(G)⊗m)rig extendsto (Lie(G)⊗m)

rig` and makes (Lie(G)⊗m)

rig` a topological Banach module over the K-Banach algebra

Λ(Γ2ε−1)K,`. The action of Γ on (Lie(G)⊗m)rig extends to a jointly continuous action of the K-Fréchetalgebra D(Γ ). The action of Γ on the strong continuous K-linear dual [(Lie(G)⊗m)rig]′b of (Lie(G)⊗m)rig islocally analytic.


Proof. Set Mm` := (Lie(G)⊗m)

rig` . Any R-basis (δ1, . . . ,δs) of Lie(G)⊗m can be viewed as an Rrig

` -basis ofMm

` . Writing Mm` =⊕s

i=1Rrig` δi, the topology of Mm

` is defined by the norm

||s

∑i=1

fiδi||` = supi|| fi||` if f1, . . . , fs ∈ Rrig

` .

We choose an ordered basis (γ1, . . . ,γr) of Γ2ε−1 and let bi := γi− 1 be as before. By induction on |α| wewill first prove the fundamental estimate ||bα δ ||` ≤ ||bα ||`||δ ||` for all α ∈ Nr and δ ∈ (Lie(G)⊗m)rig. Asin the proof of Theorem 6 this is reduced to the case |α| = 1, i.e. bα = γi− 1 for some 1 ≤ i ≤ r. Further,we may assume δ = f δ j for some f ∈ Rrig and 1≤ j ≤ s.

There are elements r1, . . . ,rs ∈ R such that γi(δ j) = ∑sν=1 rν δν . According to Theorem 5 we have (γi−

1)(δ j) ∈ mε Lie(G)⊗m, i.e. r j−1 ∈ mε and rν ∈ mε for ν 6= j. We claim that ||r||` ≤ p−c/` for any integerc≥ 0 and any element r ∈mc. Indeed, this is clear for c = 0. For general c, the ideal mc of R is generated byall elements of the form pauβ with a ∈ N, β ∈ N(h−d)d and a+ |β |= c. Since `≥ 1 we have |pa|= p−a ≤p−a/`, and the claim follows from the multiplicativity of the norm || · ||` on R. Now

||(γi−1)( f δ j)||` ≤ max||(γi−1)( f ) · γi(δ j)||`, || f · (γi−1)(δ j)||`= max||∑

ν

(γi−1)( f )rν δν ||`, || f ||`||δ j−∑ν

rν δν ||`,

where ||(γi− 1)( f ) · rν ||` ≤ ||(γi− 1)( f )||` ≤ p(2ε−1)/`|| f ||` by Proposition 2. Here p(2ε−1)/` ≤ p−ε/` =||γi− 1||`. Moreover, ||r j− 1||` ≤ p−ε/` and ||rν ||` ≤ p−ε/` if ν 6= j by the above claim. This finishes theproof of the fundamental estimate.

As an immediate consequence, we obtain that ||λ · δ ||` ≤ ||λ ||`||δ ||` for any λ ∈ Λ(Γ2ε−1)K and anyδ ∈ Lie(G)⊗m⊗W K. The proof proceeds now as in Theorem 6.

According to [24], Theorem 4.10, the projective system (Λ(Γ2ε−1)K,`)` of K-Banach algebras en-dow their projective limit D(Γ2ε−1) with the structure of a K-Fréchet-Stein algebra. In the terminologyof [24], section 8, the family ((Lie(G)⊗m)

rig` )` is a sheaf over (D(Γ2ε−1),(|| · ||`)`) with global sections

(Lie(G)⊗m)rig for any integer m. One of the main open questions in this setting is whether this sheaf is co-herent, i.e. whether the Λ(Γ2ε−1)K,`-modules (Lie(G)⊗m)

rig` are finitely generated and whether the natural

mapsΛ(Γ2ε−1)K,`⊗Λ(Γ2ε−1)K,`+1

(Lie(G)⊗m)rig`+1 −→ (Lie(G)⊗m)

rig`

are always bijective. This would amount to the admissibility of the locally analytic Γ -representation[(Lie(G)⊗m)rig]′b in the sense of [24], section 6. Nothing in this direction is known. In the next section,however, we will have a closer look at the case dim(G) = 1 and ` = 1. We will see that in order to obtainfinitely generated objects, one might be forced to introduce yet another type of Banach algebras.

4 Non-commutative divided power envelopes

In this final section we assume that our fixed p-divisible formal group G over the algebraically closed fieldk of characteristic p is of dimension one. If h denotes the height of G then the endomorphism ring of G isisomorphic to the maximal order oD of the central Qp-division algebra D of invariant 1

h +Z (cf. [9], Propo-sition 13.10). In the following we will identify End(G) and oD (resp. Aut(G) and o∗D). We will also excludethe trivial case h= 1. We continue to denote by R=Rdef

G the universal deformation ring of G (cf. Theorem 3).

Consider the period morphism Φ : Spf(R)rig → Ph−1K of Gross and Hopkins, where Ph−1

K denotes therigid analytic projective space of dimension h− 1 over K (cf. [9], section 23). In projective coordinates Φ

can be defined by Φ(x) = [ϕ0(x) : . . . : ϕh−1(x)] where ϕ0, . . . ,ϕh−1 ∈ Rrig are certain global rigid analyticfunctions on Spf(R)rig without any common zero. The power series expansions of the functions ϕi in suit-able coordinates u1, . . . ,uh−1 can be written down explicitly by means of a closed formula of Yu (cf. [13],

222 Jan Kohlhaase

Proposition 1.5 and Remark 1.6). According to [9], Lemma 23.14, the function ϕ0 does not have any zeroeson Bh−1

1 ⊂ Spf(R)rig, hence is a unit in Rrig1 . We set

wi :=ϕi

ϕ0∈ Rrig

1 for 1≤ i≤ h−1.

By [9], Lemma 23.14, any element f ∈ Rrig1 admits a unique expansion of the form f = ∑α∈Nh−1 dα wα with

dα ∈ K and lim|α|→∞ |dα |p−|α| = 0. Further, Φ restricts to an isomorphism Φ : Bh−11 → Φ(Bh−1

1 ) (cf. [9],Corollary 23.15).

Denote by Qph the unramified extension of degree h of Qp and by Zph its valuation ring. It was shownby Devinatz, Gross and Hopkins, that there exists an explicit closed embedding o∗D → GLh(Qph) of Liegroups over Qp such that Φ is o∗D-equivariant (cf. [13], Proposition 1.3 and Remark 1.4). Here o∗D acts onSpf(R)rig through the identification o∗D ' Aut(G), and it acts by fractional linear transformations on Ph−1

Kvia the embedding o∗D → GLh(Qph).

The morphism Φ is constructed in such a way that Φ∗OPh−1K

(1) = Lie(G)rig. It follows from general

properties of the inverse image functor that Φ∗OPh−1K

(m) = (Lie(G)⊗m)rig for any integer m. Restricting

to Bh−11 , we obtain an o∗D-equivariant and Rrig

1 -linear isomorphism (Lie(G)⊗m)rig1 ' Rrig

1 ·ϕm0 of free Rrig

1 -modules of rank one.

We denote by d the Lie algebra of the Lie group o∗D over Qp. It is isomorphic to the Lie algebra associatedwith the associative Qp-algebra D. According to [23], page 450, the universal enveloping algebra UK(d) :=U(d⊗Qp K) of d over K embeds into the locally analytic distribution algebra D(Γ2ε−1). Together with thenatural map D(Γ2ε−1)→Λ(Γ2ε−1)K,1, Theorem 7 allows us to view

Mm1 := (Lie(G)⊗m)

rig1

as a module over UK(d) 'U(g⊗Qph K) =: UK(g), where g := d⊗Qp Qph ' glh as Lie algebras over Qph .Explicitly, the action of an element x ∈ g on Mm

1 is given by

x(δ ) =ddt(exp(tx)(δ ))|t=0.

Here exp : g // GLh(Qph) is the usual exponential map which is defined locally around zero in g.

Further, a sufficiently small open subgroup of GLh(Qph) acts on Mm1 through the isomorphism Mm

1 'OPh−1

K(m)(Φ(Bh−1

1 )). Writing an element x ∈ g as a matrix x = (ars)0≤r,s≤h−1 with coefficients ars ∈ Qph ,fix indices 0≤ i, j ≤ h−1 and denote by xi j the matrix with entry 1 at the place (i, j) and zero everywhereelse. In the following we will formally put w0 := 1.

Lemma 3. Let i, j and m be integers with 0≤ i, j ≤ h−1. If f ∈ Rrig1 then

xi j( f ϕm0 ) =

wi

∂ f∂w j

ϕm0 , if j 6= 0,

(m f −∑h−1`=1 w`

∂ f∂w`

)ϕm0 , if i = j = 0,

wi(m f −∑h−1`=1 w`

∂ f∂w`

)ϕm0 , if i > j = 0.

Proof. If i = j and if t is sufficiently close to zero in Qph then exp(txii) is the diagonal matrix with entryexp(t) at the place (i, i) and 1 everywhere else on the diagonal. Recall that GLh(Qph) acts by fractional lin-ear transformations on the projective coordinates ϕ0, . . . ,ϕh−1 of Ph−1

K . Thus, exp(txii)(w`) =w` if ` 6= i 6= 0,exp(txii)(wi) = exp(t)wi if i 6= 0, and exp(tx00)(w`) =

1exp(t)w` for all 1≤ `≤ h−1.

If i 6= j then exp(txi j) = 1+ txi j in GLh(Qph). Thus, exp(txi j)(w`) = w` if ` 6= j 6= 0, exp(txi j)(w j) =w j + twi if j 6= 0, and exp(txi0)(w`) = w`/(1+ twi) for all 1 ≤ ` ≤ h−1. Writing f = f (w1, . . . ,wh−1) we


haveexp(txi j)( f ϕ

m0 ) = f (exp(txi j)(w1), . . . ,exp(txi j)(wh−1)) · exp(txi j)(ϕ0)

m.

Here exp(txi j)(ϕ0) = ϕ0 if j 6= 0, exp(tx00)(ϕ0) = exp(t)ϕ0 and exp(txi0)(ϕ0) = ϕ0 + tϕi if 1≤ i≤ h−1. Itis now an exercise in elementary calculus to derive the desired formulae.

Note that (Lie(G)⊗m)rig is a D(Γ2ε−1)-stable K-subspace of Mm1 and hence is g-stable. If m = 0 then

Lemma 3 shows that in order to describe the g-action in the coordinates u1, . . . ,uh−1, one essentially has tocompute the functional matrix

F := (∂ui

∂w j)1≤i, j≤h−1.

Proposition 3. The matrix A := ( ∂ϕi∂u j

ϕ0− ∂ϕ0∂u j

ϕi)1≤i, j≤h−1 over Rrig is invertible over the localization Rrigϕ0 .

We have F = ϕ20 A−1, which is a matrix with entries in ϕ0Rrig. Moreover, we have ∑

h−1j=1 ϕ j

∂ui∂w j∈ ϕ2

0 Rrig forany index 1≤ i≤ h−1.

Proof. Let B := ( ∂ϕi∂u j

)0≤i, j≤h−1 with ∂ϕi∂u0

:= ϕi. We have B ∈ GLh(Rrig) by a result of Gross and Hopkins(cf. [9], Corollary 21.17). Setting

N :=

1 0 · · · 0−ϕ1 ϕ0 0

.... . .

−ϕh−1 0 ϕ0

, we have NB =

ϕ0

∂ϕ0∂u1· · · ∂ϕ0

∂uh−10... A0

.

This already shows that A is invertible over Rrigϕ0 . Denoting by c0, . . . ,ch−1 the columns of B−1 = (ci j)i, j ∈

GLh(Rrig), we obtain

(ϕ−10

h−1

∑j=0

ϕ jc j,ϕ−10 c1, . . . ,ϕ

−10 ch−1) = B−1N−1 =

ϕ−10 ∗ · · · ∗0... A−1

0

.

By the chain rain rule we have

δi j =∂wi

∂w j=

h−1

∑`=1

∂wi

∂u`· ∂u`

∂w j=

h−1

∑`=1

ϕ−20 (

∂ϕi

∂u`ϕ0−

∂ϕ0

∂u`ϕi)

∂u`∂w j

,

so that F = ϕ20 A−1. As seen above, the right hand side has entries in ϕ0Rrig. Further, we have ∑

h−1j=1 ϕ j

∂ui∂w j

=

∑h−1j=1 ϕ jϕ0ci j =−ϕ2

0 ci0 ∈ ϕ20 Rrig for any index 1≤ i≤ h−1.

Together with Lemma 3, Proposition 3 shows that x(ui) ∈ Rrig for any x ∈ g and any 1≤ i≤ h−1, as wasclear a priori. For h = 2, Lemma 3 and Proposition 3 reprove [9], formula (25.14).

Coming back to the g-module Mm1 for general m, consider the subalgebra slh of g over Qph . Let t denote

the Cartan subalgebra of diagonal matrices in slh, and let ε1, . . . ,εh−1 ⊂ t∗ denote the basis of the rootsystem of (slh, t) given by εi(diag(t0, . . . , th−1)) := ti−1− ti. We let λ1 ∈ t∗ denote the fundamental dominantweight defined by λ1 := 1

h ∑h−1i=1 (h− i)εi. We have

λ1(diag(t0, . . . , th−1)) =1h

h−1

∑i=1

(h− i)(ti−1− ti) =1h((h−1)t0−

h−1

∑i=1

ti) = t0

for any element diag(t0, . . . , th−1) ∈ t⊂ slh.

Proposition 4. For any integer m ≥ 0, the subspace W := ∑|α|≤m K ·wα ϕm0 of Mm

1 is g-stable. The actionof slh on W is irreducible. More precisely, W is the irreducible slh-representation of highest weight m ·λ1.

224 Jan Kohlhaase

Proof. It follows from Lemma 3 that W is stable under any element xi j with j 6= 0 or i = j = 0. If 1≤ i≤h−1 and if n is a non-negative integer then

xni0(w

αϕ

m0 ) = [

n−1

∏`=0

(m−|α|− `)] ·wα wni ϕ

m0 ,

as follows from Lemma 3 by induction. Therefore, xi0(wα ϕm0 ) = 0 if |α| = m. If |α| < m then xi0(wα) has

degree |α|+1≤ m. This proves that W is g-stable.

The above formula also shows that W is generated by ϕm0 as an slh-representation. If f ϕm

0 ∈W is non-zero, then Lemma 3 shows that (xα1

01 · · ·xαh−10(h−1))( f ϕm

0 ) is a non-zero scalar multiple of ϕm0 for a suitable

multi-index α . Therefore, the slh-representation W is irreducible.

Finally, if x= diag(t0, . . . , th−1)∈ t then x(wα ϕm0 ) = (t0(m−|α|)+∑

h−1i=1 αiti) ·wα ϕm

0 by Lemma 3. Here,

t0(m−|α|)+h−1

∑i=1

αiti = t0m+h−1

∑i=1

αi(ti− t0) = (m ·λ1−h−1

∑i=1

αi

i

∑`=1

ε`)(x).

This shows that m ·λ1 is the highest weight of the slh-representation W .

Remark 2. The statement of Proposition 4 can be deduced from a stronger result of Gross and Hopkins.Namely, if m = 1 then Lie(G)rig contains an h-dimensional algebraic representation of o∗D (cf. [9], Propo-sition 23.2). Under the restriction map Lie(G)rig→ Lie(G)

rig1 , the derived representation of g= d⊗Qp Qph

maps isomorphically to the g-representation W above.

We will now see that the action of g on Mm1 naturally extends to a certain divided power completion

of the universal enveloping algebra UK(g). Note that if i, j,r and s are indices between 0 and h− 1, thenxi j · xrs = δ jrxis in g' glh. Therefore,

[xi j,xrs] = δ jrxis−δisxr j =

0, if j 6= r and i 6= s,

xis, if j = r and i 6= s,

−xr j, if j 6= r and i = s,

xii− x j j, if j = r and i = s.

Setting x′i j := pδ0i−δ0 jxi j, one readily checks that the same relations hold on replacing xi j by x′i j and xrs by x′rseverywhere. It follows that the elements x′i j span a free Zph -Lie subalgebra of g that we denote by g. Sincead(x′i j)

2 = 0 if i 6= j, and since (εi+1− ε j)([x′i j,x′i j]) = 2 if i < j, it follows from [2], VIII.12.7 Théorème 2

(iii), that the W -lattice g of g is the base extension from Z to W of a Chevalley order of g in the sense of [2],VIII.12.7 Définition 2.

For 0≤ i≤ h−1 and n≥ 0 we set(x′iin

):=

x′ii(x′ii−1) · · ·(x′ii−n+1)

n!∈UK(g).

We let U denote the W -subalgebra of UK(g) generated by the elements (x′i j)n/n! for i 6= j and n≥ 0, as well

as by the elements(x′ii

n

)for 0 ≤ i ≤ h−1 and n ≥ 0. It follows from [2], VIII.7.12 Théorème 3, that U is a

free W -module and that a W -basis of U is given by the elements

b`mn := (∏i< j

(x′i j)ì j

ì j!) · (

h−1

∏i=0

(x′iimi

)) · (∏

i> j

(x′i j)ni j

ni j!)

with ` = (ì j),n = (ni j) ∈ Nh(h−1)/2 and m = (mi) ∈ Nh. Here the products of the x′i j for i < j and i > jhave to be taken in a fixed but arbitrary ordering of the factors. For split semisimple Lie algebras these


constructions and statements are due to Kostant (cf. [14], Theorem 1, where U is denoted by B).

We denote by U the p-adic completion of the ring U and set

UdpK (g) := U⊗W K.

According to the above freeness result, any element of UdpK (g) can be written uniquely in the form

∑`,m,n d`mnb`mn with coefficients d`mn ∈ K satisfying d`mn → 0 as |`|+ |m|+ |n| → ∞. Therefore, UdpK (g)

is a K-algebra containing UK(g). We view it as a K-Banach algebra with unit ball U and call it the completedivided power enveloping algebra of g.

Theorem 8. For any integer m the action of g on (Lie(G)⊗m)rig1 extends to a continuous action of Udp

K (g).

Proof. The ring of continuous K-linear endomorphisms of Mm1 = (Lie(G)⊗m)

rig1 is a K-Banach algebra for

the operator norm. Since the latter is submultiplicative, the set of endomorphisms with operator norm lessthan or equal to one is a p-adically separated and complete W -algebra. Therefore, it suffices to prove thatany element of the form (x′i j)

n/n!, i 6= j, or(x′ii

n

), 0≤ i≤ h−1, has operator norm less than or equal to one

on Mm1 whenever n≥ 0. If α ∈ Nh−1 and 0≤ i, j ≤ h−1 then

xni j(w

αϕ

m0 ) =

αnj wα ϕm

0 , if i = j 6= 0,

(m−|α|)nwα ϕm0 , if i = j = 0,

n!(

α jn

)wα w−n

j wni ϕm

0 , if i 6= j 6= 0,

n!(m−|α|

n

)wα wn

i ϕm0 , if i 6= j = 0,

(1)

as follows from Lemma 3 by induction. Here the generalized binomial coefficients are defined by(xn

):=

x(x−1) · · ·(x−n+1)n!

∈ Z

for any integer x. Now ||(∑α dα wα)ϕm0 ||1 = supα|dα |p−|α|. Bearing in mind our convention w0 = 1, we

obtain the claim for (x′i j)n/n! if i 6= j. If 0≤ i≤ h−1 then we obtain

(x′iin

)(wα

ϕm0 ) =

(αin

)wα ϕm

0 , if i 6= 0,(m−|α|n

)wα ϕm

0 , if i = 0.

This completes the proof.

Theorem 9. Let m be an integer and set c :=wmax−1,m+11 ϕm

0 . The U(g)-submodule U(g)·c of (Lie(G)⊗m)rig1

is dense. If h = 2 and m≥−1 then UdpK (g) · c = (Lie(G)⊗m)

rig1 .

Proof. Equation (1) shows that xmax−1,m+101 xα1

10 · · ·xαh−1(h−1)0 · c is a non-zero scalar multiple of wα ϕm

0 . Thus,K[w] ·ϕm

0 ⊂UK(g) · c, proving the first assertion.

If h= 2 and m≥−1 let us be more precise. Setting m′ :=max−1,m+1, w :=w1 and x := x′10, we havexn · c = (−1)nn!p−nwn+m′ϕm

0 for any n≥ 0 because(−1

n

)= (−1)n. If f = ∑n≥0 dnwn ∈ Rrig

1 then dn pn→ 0in K. Therefore, λ := ∑n≥0 dn+m′(−p)n xn

n! converges in UdpK (g) and we have f ϕm

0 −λ · c = ∑m′−1n=0 dnwnϕm

0 .The latter is contained in K[w] ·ϕm

0 ⊂UK(g) · c, as seen above.

Remark 3. By a result of Lazard, the image of UK(g) 'UK(d) in Λ(Γ2ε−1)K,1 is dense (cf. [15], ChapitreIV, Théorème 3.2.5). We state without proof that the completion of UK(g) for the norm || · ||1 embeds

continuously into UdpK (g). However, a formal series like ∑n≥0 pn (x′10)

n

n! = ∑n≥0xn

10n! does not converge in

Λ(Γ2ε−1)K,1. Therefore, one might have doubts whether Mm1 is still finitely generated over Λ(Γ2ε−1)K,1.

226 Jan Kohlhaase

References

1. S. BOSCH, U. GÜNTZER, R. REMMERT: Non-Archimedean Analysis, Grundlehren Math. Wiss. 261, Springer, 19842. N. BOURBAKI: Groupes et Algèbres de Lie, Chapitres 7 et 8, Springer, 20063. N. BOURBAKI: Algèbre Commutative, Chapitres 1 à 4, Springer, 20064. A. BRUMER: Pseudocompact Algebras, Profinite Groups and Class formations, J. Algebra 4, 1966, pp. 442–4705. P. CARTIER: Relèvements des groupes formels commutatifs, Séminaire Bourbaki 359, 21e année, 1968/696. M. DEMAZURE: Lectures on p-divisible groups, Lecture Notes in Mathematics 302, 19867. E.S. DEVINATZ, M.J. HOPKINS: The action of the Morava stabilizer group on the Lubin-Tate moduli space of lifts,

Amer. J. Math. 117, No. 3, 1995, pp. 669–7108. J.D. DIXON, M.P.F. DU SAUTOY, A. MANN, D. SEGAL: Analytic Pro-p Groups, 2nd Edition, Cambridge Studies in

Advanced Mathematics 61, Cambridge University Press, 20039. B.H. GROSS, M.J. HOPKINS: Equivariant vector bundles on the Lubin-Tate moduli space, Contemporary Math. 158,

1994, pp. 23–8810. J. DE JONG: Crystalline Dieudonné module theory via formal and rigid geometry, Publ. IHES 82, 1995, pp. 5–9611. M. HAZEWINKEL: Formal Groups and Applications, Pure and Applied Mathematics 78, Academic Press, 197812. J. KOHLHAASE, B. SCHRAEN: Homological vanishing theorems for locally analytic representations, Math. Ann. 353,

2012, pp. 219–25813. J. KOHLHAASE: On the Iwasawa theory of the Lubin-Tate moduli space, Comp. Math. 149, 2013, pp. 793–83914. B. KOSTANT: Groups over Z, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder,

Colorado, 1965), AMS, 1966, pp. 90–9815. M. LAZARD: Groupes analytiques p-adiques, Publ. I.H.É.S. 26, 1965, pp. 5–21916. M. LAZARD: Commutative Formal Groups, Lecture Notes in Mathematics 443, 197517. J. LUBIN, J. TATE: Formal moduli for one-parameter formal Lie groups, Bulletin de la S.M.F., tome 94, 1966, pp. 49–5918. S. ORLIK: Equivariant vector bundles on Drinfeld’s upper half space, Invent. Math. 172, 2008, pp. 585–65619. M. RAPOPORT, TH. ZINK: Period spaces for p-divisible groups, Annals of Math. Studies 141, Princeton Univ. Press,

199620. T. SCHMIDT: Auslander regularity of p-adic distribution algebras, Representation Theory 12, 2008, pp. 37–5721. P. SCHNEIDER: Nonarchimedean Functional Analysis, Springer Monographs in Mathematics, Springer, 200222. P. SCHNEIDER: p-Adic Lie Groups, Grundlehren Math. Wiss. 344, Springer, 201123. P. SCHNEIDER, J. TEITELBAUM: Locally analytic distributions and p-adic representation theory, with applications to

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26. J. TATE: p-divisible Groups, Proc. Conf. Local Fields (Driebergen, 1966), Springer, 1967, pp. 158–18327. H. UMEMURA: Formal moduli for p-divisible formal groups, Nagoya Math. J. 42, 1977, pp. 1–728. TH. ZINK: Cartiertheorie kommutativer formaler Gruppen, Teubner-Texte zur Mathematik 68, Leipzig, 1984

The structure of Selmer groups of elliptic curves and modularsymbols

Masato Kurihara

Abstract For an elliptic curve over the rational number field and a prime number p, we study the structureof the classical Selmer group of p-power torsion points. In our previous paper [12], assuming the mainconjecture and the non-degeneracy of the p-adic height pairing, we proved that the structure of the Selmergroup with respect to p-power torsion points is determined by some analytic elements δm defined frommodular symbols (see Theorem 1.1.1 below). In this paper, we do not assume the main conjecture nor thenon-degeneracy of the p-adic height pairing, and study the structure of Selmer groups (see Theorems 1.2.3and 1.2.5), using these analytic elements and Kolyvagin systems of Gauss sum type.

1 Introduction

1.1 Structure theorem of Selmer groups

Let E be an elliptic curve over Q. Iwasawa theory, especially the main conjecture gives a formula on theorder of the Tate Shafarevich group by using the p-adic L-function (cf. Schneider [24]). In this paper, as asequel of [10], [11] and [12], we show that we can derive more information than the order, on the structureof the Selmer group and the Tate Shafarevich group from analytic quantities, in the setting of our paper,from modular symbols.

In this paper, we consider a prime number p such that(i) p is a good ordinary prime > 2 for E,(ii) the action of GQ on the Tate module Tp(E) is surjective where GQ is the absolute Galois group of Q,(iii) the (algebraic) µ-invariant of (E,Q∞/Q) is zero where Q∞/Q is the cyclotomic Zp-extension, namelythe Selmer group Sel(E/Q∞,E[p∞]) (for the definition, see below) is a cofinitely generated Zp-module,(iv) p does not divide the Tamagawa factor Tam(E)=Π`:bad(E(Q`) : E0(Q`)), and p does not divide #E(Fp)(namely not anomalous).We note that the property (iii) is a conjecture of Greenberg since we are assuming (ii).

For a positive integer N > 0, we denote by E[pN ] the Galois module of pN-torsion points, and E[p∞] =⋃N>0 E[pN ]. For an algebraic extension F/Q, Sel(E/F,E[pN ]) is the classical Selmer group defined by

Sel(E/F,E[pN ]) = Ker(H1(F,E[pN ])−→∏v

H1(Fv,E[pN ])/E(Fv)⊗Z/pN),

so Sel(E/F,E[pN ]) fits into an exact sequence

0−→ E(F)⊗Z/pN −→ Sel(E/F,E[pN ])−→X(E/F)[pN ]−→ 0

where X(E/F) is the Tate Shafarevich group over F . We define Sel(E/F,E[p∞]) = lim−→

Sel(E/F,E[pN ]).

Department of Mathematics, Keio University

227

228 Masato Kurihara

Let P(N) be the set of prime numbers ` such that ` is a good reduction prime for E and `≡ 1 (mod pN).

For each `, we fix a generator η` of (Z/`Z)× and define logF`(a) ∈ Z/(`−1) by η

logF`(a)

` ≡ a (mod `).Let f (z)=Σane2πinz be the modular form corresponding to E. For a positive integer m and the cyclotomic

field Q(µm), we denote by σa ∈ Gal(Q(µm)/Q) the element such that σa(ζ ) = ζ a for any ζ ∈ µm. Weconsider the modular element ∑

ma=1,(a,m)=1[

am ]σa ∈C[Gal(Q(µm)/Q)] of Mazur and Tate ([16]) where [ a

m ] =

2πi∫ a/m

∞f (z)dz is the usual modular symbol. We only consider the real part

θQ(µm) =m

∑a=1

(a,m)=1

Re([ am ])

Ω+E

σa ∈Q[Gal(Q(µm)/Q)] (1)

where Ω+E =

∫E(R) ωE is the Néron period. Suppose that m is a squarefree product of primes in P(N). Since

we are assuming the GQ-module E[p] of p-torsion points is irreducible, we know θQ(µm) ∈Zp[Gal(Q(µm)/Q)]

(cf. [27]). We consider the coefficient of θQ(µm) of “∏`|m(ση`−1)", more explicitly we define

δm =m

∑a=1

(a,m)=1

Re([ am ])

Ω+E

(∏`|m

logF`(a)) ∈ Z/pN (2)

where logF`(a) means the image of logF`

(a) under the canonical homomorphism Z/(`−1)−→ Z/pN . Letordp : Z/pN −→ 0,1, ...,N− 1,∞ be the p-adic valuation normalized as ordp(p) = 1 and ordp(0) = ∞.We note that ordp(δm) does not depend on the choices of η` for `|m. We define δ1 = θQ = Re([0])/Ω

+E =

L(E,1)/Ω+E .

For a squarefree product m of primes, we define ε(m) to be the number of prime divisors of m, namelyε(m) = r if m = `1 · ... · `r. Let N(N) be the set of squarefree products of primes in P(N). We suppose 1 isin N(N). For each integer i ≥ 0, we define the ideal Θi(Q)(N,δ ) of Z/pN to be the ideal generated by all δmsuch that ε(m)≤ i for all m ∈N(N);

Θi(Q)(N,δ ) = (δm | ε(m)≤ i and m ∈N(N))⊂ Z/pN . (3)

We define ni,N ∈ 0,1, ...,N−1,∞ by Θi(Q)(N,δ ) = pni,N (Z/pN) (we define ni,N = ∞ if Θi(Q)(N,δ ) = 0).

Theorem 1.1.1 ([12] Theorem B, Theorem 9.3.1 and (9.14)) We assume that the main conjecture for(E,Q∞/Q) (see (1)) and the p-adic height pairing is non-degenerate.(1) ni,N does not depend on N when N is sufficiently large (for example, when N > 2ordp(η0) where η0 isthe leading term of the p-adic L-function, see §9.4 in [12]). We put ni = ni,N for N 0. In other words, wedefine ni by

lim←−

Θi(Q)(N,δ ) = pniZp ⊂ Zp.

We denote this ideal of Zp by Θi(Q)(δ ).(2) Consider the Pontrjagin dual Sel(E/Q,E[p∞])∨ of the Selmer group. Suppose that

rankZp Sel(E/Q,E[p∞])∨ = r(∈ Z≥0), and dimFp Sel(E/Q,E[p])∨ = a.

Then we haveΘ0(Q)(δ ) = ...=Θr−1(Q)(δ ) = 0 and Θr(Q)(δ ) 6= 0.

For any i≥ r, ni is an even number, and

pnr = #(Sel(E/Q,E[p∞])∨)tors,

na = 0, and

Sel(E/Q,E[p∞])∨ ' Z⊕rp ⊕ (Z/p

nr−nr+22 )⊕2⊕ (Z/p

nr+2−nr+42 )⊕2⊕ ...⊕ (Z/p

na−2−na2 )⊕2

hold.

In particular, knowing Θi(Q)(δ ) for all i ≥ 0 completely determines the structure of Sel(E/Q,E[p∞])∨

as a Zp-module. Namely, the modular symbols determine the structure of the Selmer group under ourassumptions.

The structure of Selmer groups of elliptic curves and modular symbols 229

1.2 Main Results

We defineP(N)1 = ` ∈ P(N) | H0(F`,E[pN ])' Z/pN.

This is an infinite set by Chebotarev density theorem since we are assuming (ii) (see [12] §4.3). We defineN

(N)1 to be the set of squarefree products of primes in P

(N)1 . Again, we suppose 1 ∈ N

(N)1 . We propose the

following conjecture.

Conjecture 1.2.1 There is m ∈N(N)1 such that δm is a unit in Z/pN , namely

ordp(δm) = 0.

Numerically, it is easy to compute δm, so it is easy to check this conjecture.

Theorem 1.2.2 ([12] Theorem 9.3.1) If we assume the main conjecture and the non-degeneracy of thep-adic height pairing, Conjecture 1.2.1 holds true.

In fact, we obtain Conjecture 1.2.1, considering the case i = s in Theorem 9.3.1 in [12].

From now on, we do not assume the main conjecture (1) nor the non-degeneracy of the p-adic heightpairing.

We define the Selmer group Sel(Z[1/m],E[pN ]) by

Sel(Z[1/m],E[pN ]) = Ker(H1(Q,E[pN ])−→∏v6 |m

H1(Qv,E[pN ])/E(Qv)⊗Z/pN).

If all bad primes and p divide m, we know Sel(Z[1/m],E[pN ]) is equal to the étale cohomology groupH1

et(SpecZ[1/m],E[pN ]), which explains the notation “Sel(Z[1/m],E[pN ])". (We use Sel(Z[1/m],E[pN ])

for m ∈N(N)1 in this paper, but E[pN ] is not an étale sheaf on SpecZ[1/m] for such m.)

Let λ be the λ -invariant of Sel(E/Q∞,E[p∞])∨. We put nλ = minn∈Z | pn−1≥ λ and dn = nλ +Nnfor n ∈ Z≥0. We define

P(N,n)1 = ` ∈ P

(N)1 | `≡ 1 (mod pdn ) (1)

(then P(N,n)1 ⊂ P

(N)1 (Q[n]) holds, see the end of §3.1 for this fact, and see §3.1 for the definition of the set

P(N)1 (Q[n])). We denote by N

(N,n)1 the set of squarefree products of primes in P

(N,n)1 .

In this paper, for any finite abelian p-extension K/Q in which all bad primes of E are unramified, weprove in §4 the following theorem for Z/pN [Gal(K/Q)]-modules Sel(E/K,E[pN ]) and Sel(OK [1/m],E[pN ])(see Corollary 4.1.3 and Theorem 4.2.1). We simply state it in the case K = Q below. An essentialingredient in this paper is the Kolyvagin system of Gauss sum type. We construct Kolyvagin systemsκm,` ∈ Sel(Z[1/m`],E[pN ]) for (m, `) satisfying `∈P(N,ε(m`)+1)

1 and m`∈N(N,ε(m`)+1)1 (see §3.4 and Propo-

sitions 3.4.2) by the method in [12]. (We can construct these elements, using the half of the main conjectureproved by Kato [7].) The essential difference between our Kolyvagin systems κm,` of Gauss sum type andKolyvagin systems in Mazur and Rubin [14] is that our κm,` is related to L-values. In particular, κm,` sat-isfies a remarkable property φ`(κm,`) = −δm`t`,K (see Propositions 3.4.2 (4)) though we do not explain thenotation here.

Theorem 1.2.3 Assume that ordp(δm) = 0 for some m ∈N(N)1 .

(1) The canonical homomorphism

sm : Sel(E/Q,E[pN ])−→⊕`|m

E(Q`)⊗Z/pN '⊕`|m

E(Q`)⊗Z/pN ' (Z/pN)ε(m)

is injective.(2) Assume further that m ∈N

(N,ε(m)+1))1 and that m is admissible (for the definition of the notion “admissi-

ble", see the paragraph before Proposition 3.3.2). Then Sel(Z[1/m],E[pN ]) is a free Z/pN-module of rankε(m), and κ m

` ,``|m is a basis of Sel(Z[1/m],E[pN ]).

(3) We define a matrix A as in (1) in Theorem 4.2.1, using κ m` ,`

. Then A is a relation matrix of the Pontrjagin

dual Sel(E/Q,E[pN ])∨ of the Selmer group; namely if fA : (Z/pN)ε(m) −→ (Z/pN)ε(m) is the homomor-phism corresponding to the above matrix A, then we have

230 Masato Kurihara

Coker( fA)' Sel(E/Q,E[pN ])∨.

It is worth noting that we get nontrivial (moreover, linearly independent) elements in the Selmer groups.

The ideals Θi(Q)(δ ) in Theorem 1.1.1 are not suitable for numerical computations because we have tocompute infinitely many δm. On the other hand, we can easily find m with ordp(δm) = 0 numerically. Sincesm is injective, we can get information of the Selmer group from the image of sm, which is an advantage ofTheorem 1.2.3 and the next Theorem 1.2.5 (see also the comment in the end of Example (5) in §5.3).

We next consider the case N = 1, so Sel(E/Q,E[p]). Now we regard δm as an element of Fp for m∈N(1)1 .

We say m is δ -minimal if δm 6= 0 and δd = 0 for all divisors d of m with 1 ≤ d < m. Our next conjectureclaims that the structure (the dimension) of Sel(E/Q,E[p]) is determined by a δ -minimal m, therefore canbe easily computed numerically.

Conjecture 1.2.4 If m ∈N(1)1 is δ -minimal, the canonical homomorphism

sm : Sel(E/Q,E[p])−→⊕`|m

E(Q`)⊗Z/p'⊕`|m

E(F`)⊗Z/p' (Z/pN)ε(m)

is bijective. In particular, dimFp Sel(E/Q,E[p]) = ε(m).

If m ∈N(1)1 is δ -minimal, the above homomorphism sm : Sel(E/Q,E[p])−→ (Z/pN)ε(m) is injective by

Theorem 1.2.3 (1), so we knowdimFp Sel(E/Q,E[p])≤ ε(m).

Therefore, the problem is in showing the other inequality.We note that the analogue of the above conjecture for ideal class groups does not hold (see §5.4). But

we hope that Conjecture 1.2.4 holds for the Selmer groups of elliptic curves. We construct in §5 a modifiedversion κ

q,q′,zm,` of Kolyvagin systems of Gauss sum type for any (m, `) with m` ∈ N

(N)1 . (The Kolyvagin

system κm,` in §3 is defined for (m, `) with m` ∈N(N,ε(m`)+1)1 , but κ

q,q′,zm,` is defined for more general (m, `),

namely for (m, `) with m` ∈N(N)1 .) Using the modified Kolyvagin system κ

q,q′,zm,` , we prove the following.

Theorem 1.2.5 (1) If ε(m) = 0, 1, then Conjecture 1.2.4 is true.(2) If there is ` ∈ P(1) which is δ -minimal (so ε(`) = 1), then

Sel(E/Q,E[p∞])'Qp/Zp.

Moreover, if there is ` ∈ P(1)1 which is δ -minimal and which satisfies ` ≡ 1 (mod pn

λ ′+2) where λ ′ is theanalytic λ -invariant of (E,Q∞/Q), then the main conjecture (1) for Sel(E/Q∞,E[p∞]) holds true. In thiscase, Sel(E/Q∞,E[p∞])∨ is generated by one element as a Zp[[Gal(Q∞/Q)]]-module.(3) If ε(m) = 2 and m is admissible, then Conjecture 1.2.4 is true.(4) Suppose that ε(m) = 3 and m = `1`2`3. Assume that m is admissible and the natural maps s`i :Sel(E/Q,E[p])−→ E(F`i)⊗Z/p are surjective both for i = 1 and i = 2. Then Conjecture 1.2.4 is true.

In this way, we can determine the Selmer groups by finite numbers of computations in several cases. Wegive several numerical examples in §5.2.

Remark 1.2.6 Concerning the Fitting ideals and the annihilator ideals of some Selmer groups, we provethe following in this paper. Let K/Q be a finite abelian p-extension in which all bad primes of E are unram-ified. We take a finite set S of good reduction primes, which contains all ramifying primes in K/Q except p.Let m be the product of primes in S. We prove that the initial Fitting ideal of the RK =Zp[Gal(K/Q)]-moduleSel(OK [1/m],E[p∞])∨ is principal, and

ξK,S ∈ Fitt0,RK (Sel(OK [1/m],E[p∞])∨)

where ξK,S is an element of RK which is explicitly constructed from modular symbols (see (2)). If themain conjecture (1) for (E,Q∞/Q) holds, the equality Fitt0,RK (Sel(OK [1/m],E[p∞])∨) = ξK,SRK holds (seeRemark 2.3.2). We prove the Iwasawa theoretical version in Theorem 2.2.2.

Let ϑK be the image of the p-adic L-function, which is also explicitly constructed from modular symbols.We show in Theorem 2.3.1


ϑK ∈ AnnRK (Sel(OK [1/m],E[p∞])∨).

Concerning the higher Fitting ideals (cf. §2.4), we show

δm ∈ Fittε(m),Z/pN (Sel(E/Q,E[pN ])∨)

where Fitti,R(M) is the i-th Fitting ideal of an R-module M. We prove a slightly generalized version for Kwhich is in the cyclotomic Zp-extension Q∞ of Q (see Theorem 2.4.1 and Corollary 2.4.2).

I would like to thank John Coates heartily for his helpful advice and for discussion with him, especiallyfor the discussion in March 2013, which played an essential role in my producing this paper. I also thankheartily Kazuya Kato for his constant interest in the results of this paper. I also thank Kazuo Matsuno andChristian Wuthrich very much for their helping me to compute modular symbols.

2 Selmer groups and p-adic L-functions

2.1 Modular symbols and p-adic L-functions

Let E be an elliptic curve over Q, and f (z) = Σane2πinz the modular form corresponding to E. In thissection, we assume that p is a prime number satisfying (i), (ii), (iii) in §1.1. We define Pgood = ` | `is a good reduction prime for E \ p. For any finite abelian extension K/Q, we denote by K∞/K thecyclotomic Zp-extension. For a real abelian field K of conductor m, we define θK to be the image of θQ(µm)

in Q[Gal(K/Q)] where θQ(µm) is defined in (1).We write

RK = Zp[Gal(K/Q)] and ΛK∞= Zp[[Gal(K∞/Q)]].

For any positive integer n, we simply write RQ(µn) = Rn in this subsection. For any positive integers d,c such that d | c, we define the norm map νc,d : Rd = Zp[Gal(Q(µd)/Q)] −→ Rc = Zp[Gal(Q(µc)/Q)] byσ 7→ ∑τ where for σ ∈ Gal(Q(µd)/Q), τ runs over all elements of Gal(Q(µc)/Q) such that the restrictionof τ to Q(µd) is σ . Let m be a squarefree product of primes in Pgood , and n a positive integer. By ourassumption (ii), we know θQ(µmpn ) ∈ Rmpn (cf. [27]). Let α ∈ Z×p be the unit root of x2− apx+ p = 0 andput

ϑQ(µmpn ) = α−n(θQ(µmpn )−α

−1νmpn,mpn−1(θQ(µmpn−1 ))) ∈ Rmpn

as usual. Then ϑQ(µmpn )n≥1 is a projective system (cf. Mazur and Tate [16] the equation (4) on page 717)and we obtain an element ϑQ(µmp∞ ) ∈ ΛQ(µmp∞ ), which is the p-adic L-function of Mazur and Swinnerton-Dyer.

We also use the notation Λnp∞ = ΛQ(µnp∞ ) for simplicity. Suppose that a prime ` does not divide mp, andcm`,m : Λm`p∞ −→Λmp∞ is the natural restriction map. Then we know

cm`,m(ϑQ(µm`p∞ )) = (a`−σ`−σ−1` )ϑQ(µmp∞ ) (1)

(cf. Mazur and Tate [16] the equation (1) on page 717).We will construct a slightly modified element ξQ(µmp∞ ) in Λmp∞ . We put P′`(x) = x2−a`x+ `. Let m be a

squarefree product of Pgood . For any divisor d of m and a prime divisor ` of m/d, σ` ∈ Gal(Q(µd p∞)/Q) =lim←−

Gal(Q(µd pn)/Q) is defined as the projective limit of σ` ∈Gal(Q(µd pn)/Q). We consider P′`(σ`)∈Λd p∞ .Note that

−σ−1` = (−σ

−1` P′`(σ`)− (a`−σ`−σ

−1` ))/(`−1) ∈Λd p∞ . (2)

We put αd,m = (∏`|md (−σ−1` ))ϑQ(µd p∞ ) ∈Λd p∞ and

ξQ(µmp∞ ) = ∑d|m

νm,d(αd,m) ∈Λmp∞

where νm,d : Λd p∞ −→ Λmp∞ is the norm map defined similarly as above. (This modification ξQ(µmp∞ ) isdone by the same spirit as Greither [5] in which the Deligne-Ribet p-adic L-functions are treated.) Supposethat ` ∈ Pgood is prime to m. Then by the definition of ξQ(µmp∞ ) and (1) and (2), we have

232 Masato Kurihara

cm`,m(ξQ(µm`p∞ )) = cm`,m(∑d|m

νm`,d(αd,m`)+∑d|m

νm`,d`(αd`,m`))

= (`−1)∑d|m

νm,d(−σ−1` αd,m)+∑

d|mνm,d(cd`,d(αd`,m`))

= (`−1)∑d|m

νm,d(−σ−1` αd,m)+∑

d|mνm,d((a`−σ`−σ

−1` )αd,m)

= (−σ−1` P′`(σ`))∑

d|mνm,d(αd,m)

= (−σ−1` P′`(σ`))ξQ(µmp∞ ). (3)

We denote by ϑQ(µm) ∈ RQ(µm) the image of ϑQ(µmp∞ ) under the natural map ΛQ(µmp∞ ) −→ RQ(µm). Wehave

ϑQ(µm) = (1−σp

α)(1−

σ−1p

α)θQ(µm). (4)

Since we are assuming ap 6≡ 1 (mod p), we also have α 6≡ 1 (mod p), so (1− σpα)(1− σ−1

pα

) is a unit inRQ(µm).

2.2 Selmer groups

For any algebraic extension F/Q, we denote by OF the integral closure of Z in F . For a positive integerm > 0, we define a Selmer group Sel(OF [1/m],E[p∞]) by

Sel(OF [1/m],E[p∞]) = Ker(H1(F,E[p∞])−→∏v6 |m

H1(Fv,E[p∞])/E(Fv)⊗Qp/Zp)

where v runs over all primes of F which are prime to m. Similarly, for a positive integer N, we defineSel(OF [1/m],E[pN ]) by

Sel(OF [1/m],E[pN ]) = Ker(H1(F,E[pN ])−→∏v6 |m

H1(Fv,E[pN ])/E(Fv)⊗Z/pN).

In the case m = 1, we denote them by Sel(OF ,E[p∞]), Sel(OF ,E[pN ]), which are classical Selmer groups.We also use the notation Sel(E/F,E[p∞]), Sel(E/F,E[pN ]) for them, namely

Sel(E/F,E[p∞]) = Sel(OF ,E[p∞]), Sel(E/F,E[pN ]) = Sel(OF ,E[pN ]).

For a finite abelian extension K/Q, we denote by K∞/K the cyclotomic Zp-extension, and put ΛK∞=

Zp[[Gal(K∞/Q)]]. The Pontrjagin dual Sel(OK∞,E[p∞])∨ is a torsion ΛK∞

-module (Kato [7] Theorem 17.4).When the conductor of K is m, we define ϑK∞

∈ ΛK∞to be the image of ϑQ(µmp∞ ), and also ξK∞

∈ ΛK∞

to be the image of ξQ(µmp∞ ). The Iwasawa main conjecture for (E,Q∞/Q) is the equality between thecharacteristic ideal of the Selmer group and the ideal generated by the p-adic L-function;

char(Sel(OQ∞,E[p∞])∨) = ϑQ∞

ΛQ∞. (1)

Since we are assuming the Galois action on the Tate module is surjective, we know ϑQ∞∈ char(Sel(OQ∞

,E[p∞])∨)by Kato [7] Theorem 17.4. Skinner and Urban [26] proved the equality (1) under mild conditions. Namely,under our assumptions (i), (ii), they proved the main conjecture (1) if there is a bad prime ` which is ramifiedin Q(E[p]) ([26] Theorem 3.33).

More generally, let ψ be an even Dirichlet character and K be the abelian field corresponding to the kernelof ψ , namely K is the field such that ψ induces a faithful character of Gal(K/Q). We assume K∩Q∞ = Q.In this paper, for any finite abelian p-group G, any Zp[G]-module M and any character ψ : G −→ Q×p , wedefine the ψ-quotient Mψ by M⊗Zp[G] Oψ where Oψ = Zp[Imageψ] which is regarded as a Zp[G]-moduleby σx = ψ(σ)x for any σ ∈ G and x ∈ Oψ . We consider (Sel(OK∞

,E[p∞])∨)ψ , which is a Λψ -modulewhere Λψ = (ΛK∞

)ψ = Oψ [[Gal(K∞/K)]]. We denote the image of ϑK∞in Λψ by ψ(ϑK∞

). Then the mainconjecture states

char((Sel(OK∞,E[p∞])∨)ψ) = ψ(ϑK∞

)Λψ . (2)


We also note that ψ(ϑK∞)Λψ =ψ(ξK∞

)Λψ . By Kato [7], we know ψ(ϑK∞), ψ(ξK∞

)∈ char((Sel(OK∞,E[p∞])∨)ψ).

Let S⊂Pgood be a finite set of good primes, and K/Q be a finite abelian extension. We denote by Sram(K)the subset of S which consists of all ramifying primes in K inside S. Recall that P′`(x) = x2− a`x+ `. Wedefine

ξK∞,S = ξK∞ ∏`∈S\Sram(K)

(−σ−1` P′`(σ`)).

So ξK∞,S = ξK∞if S contains only ramifying primes in K. Suppose that S contains all ramifying primes in K

and F is a subfield of K. We denote by cK∞/F∞: ΛK∞

−→ΛF∞the natural restriction map. Using (3) and the

above definition of ξK∞,S, we havecK∞/F∞

(ξK∞,S) = ξF∞,S. (3)

For any positive integer m whose prime divisors are in Pgood , we have an exact sequence

0−→ Sel(OK∞,E[p∞])−→ Sel(OK∞

[1/m],E[p∞])−→⊕v|m

H1(K∞,v,E[p∞])−→ 0

because E(K∞,v)⊗Qp/Zp = 0 (for the surjectivity of the third map, see Greenberg Lemma 4.6 in [3]). Fora prime v of K∞, let K∞,v,nr/K∞,v be the maximal unramified extension, and Γv = Gal(K∞,v,nr/K∞,v). Supposev divides m. Since v is a good reduction prime, we have H1(K∞,v,E[p∞]) = HomCont(GK∞,v,nr ,E[p

∞])Γv =

E[p∞](−1)Γv where (−1) is the Tate twist. By the Weil pairing, the Pontrjagin dual of E[p∞](−1) is theTate module Tp(E). Therefore, taking the Pontrjagin dual of the above exact sequence, we have an exactsequence

0−→⊕v|m

Tp(E)Γv −→ Sel(OK∞[1/m],E[p∞])∨ −→ Sel(OK∞

,E[p∞])∨ −→ 0. (4)

Note that Tp(E)Γv is free over Zp because Γv is profinite of order prime to p.

Let K/Q be a finite abelian p-extension in which all bad primes of E are unramified. Suppose that S isa finite subset of Pgood such that S contains all ramifying primes in K/Q except p. Let m be a squarefreeproduct of all primes in S.

Theorem 2.2.1 (Greenberg) Sel(OK∞[1/m],E[p∞])∨ is of projective dimension ≤ 1 as a ΛK∞

-module.

This is proved by Greenberg in [4] Theorem 1 (the condition (iv) in §1.1 in this paper is not needed here,see also Proposition 3.3.1 in [4]). For more general p-adic representations, this is proved in [12] Proposition1.6.7. We will give a sketch of the proof because some results in the proof will be used later.

Since we can take some finite abelian extension K′/Q such that K∞ = K′∞ and K′ ∩Q∞ = Q, we mayassume that K ∩Q∞ = Q and p is unramified in K. Since we are assuming that E[p] is an irreducible GQ-module, we know that Sel(OK∞

,E[p∞])∨ has no nontrivial finite Zp[[Gal(K∞/K)]]-submodule by Greenberg([3] Propositions 4.14, 4.15). We also assumed that the µ-invariant of Sel(OQ∞

,E[p∞])∨ is zero, whichimplies the vanishing of the µ-invariant of Sel(OK∞

,E[p∞])∨ by Hachimori and Matsuno [6]. Therefore,Sel(OK∞

,E[p∞])∨ is a free Zp-module of finite rank. By the exact sequence (4), Sel(OK∞[1/m],E[p∞])∨ is

also a free Zp-module of finite rank.Put G = Gal(K/Q). By the definition of the Selmer group and our assumption that all primes divid-

ing m are good reduction primes, we have Sel(OK∞[1/m],E[p∞])G = Sel(OQ∞

[1/m],E[p∞]). Since we as-sumed that the µ-invariant is zero, Sel(OQ∞

[1/m],E[p∞]) is divisible. This shows that the corestriction mapSel(OK∞

[1/m],E[p∞])−→ Sel(OQ∞[1/m],E[p∞]) is surjective. Therefore, H0(G,Sel(OK∞

[1/m],E[p∞])) =0.

Next we will show that H1(G,Sel(OK∞[1/m],E[p∞])) = 0. Let NE be the conductor of E and put m′ =

mpNE . We know Sel(OK∞[1/m′],E[p∞]) is equal to the étale cohomology group H1

et(SpecOK∞[1/m′],E[p∞]).

We have an exact sequence

0−→ Sel(OK∞[1/m],E[p∞])−→ Sel(OK∞

[1/m′],E[p∞])−→⊕v|m′m

H2v (K∞,v)−→ 0 (5)

where H2v (K∞,v) = H1(K∞,v,E[p∞])/E(K∞,v)⊗Qp/Zp, and the surjectivity of the third map follows from

Greenberg Lemma 4.6 in [3]. Let E[p∞]0 be the kernel of E[p∞] = E(Q)[p∞]−→ E(Fp)[p∞] and E[p∞]et =E[p∞]/E[p∞]0. For a prime v of K∞ above p, we denote by K∞,v,nr the maximal unramified extension ofK∞,v, and put Γv = Gal(K∞,v,nr/K∞,v). We know the isomorphism H2

v (K∞,v)'−→ H1(K∞,v,nr,E[p∞]et)

Γv byGreenberg [2] §2. If v is a prime of K∞ not above p, we know H2

v (K∞,v) = H1(K∞,v,E[p∞]). Therefore, we

234 Masato Kurihara

get an isomorphism(⊕v|m′m

H2v (K∞,v))

G =⊕u|m′m

H2u (Q∞,v)

where v (resp. u) runs over all primes of K∞ (resp. Q∞) above m′/m= pNE . Thus, Sel(OK∞[1/m′],E[p∞])G−→⊕

v|m′mH2

v (K∞,v)G is surjective. On the other hand, we have H2

et(SpecOK∞[1/m′],E[p∞]) = 0 (see [2] Propo-

sitions 3, 4). This implies that

H1(G,H1et(SpecOK∞

[1/m′],E[p∞])) = H1(G,Sel(OK∞[1/m′],E[p∞])) = 0.

Taking the cohomology of the exact sequence (5), we get

H1(G,Sel(OK∞[1/m],E[p∞])) = 0. (6)

Therefore, Sel(OK∞[1/m],E[p∞]) is cohomologically trivial as a G-module by Serre [25] Chap. IX

Théorème 8. This implies that Sel(OK∞[1/m],E[p∞])∨ is also cohomologically trivial. Since Sel(OK∞

[1/m],E[p∞])∨

has no nontrivial finite submodule, the projective dimension of Sel(OK∞[1/m],E[p∞])∨ as a ΛK∞

-module is≤ 1 by Popescu [20] Proposition 2.3.

Theorem 2.2.2 Let K/Q be a finite abelian p-extension in which all bad primes of E are unramified. Wetake a finite set S of good reduction primes which contains all ramifying primes in K/Q except p. Let m bethe product of primes in S. Then(1) ξK∞,S is in the initial Fitting ideal Fitt0,ΛK∞

(Sel(OK∞[1/m],E[p∞])∨).

(2) We haveFitt0,ΛK∞

(Sel(OK∞[1/m],E[p∞])∨) = ξK∞,SΛK∞

if and only if the main conjecture (1) for (E,Q∞/Q) holds.

Proof. As we explained in the proof of Theorem 2.2.1, we may assume that K ∩Q∞ = Q. We recall thatSel(OK∞

[1/m],E[p∞])∨ is a free Zp-module of finite rank under our assumptions.

(1) Let ψ : Gal(K/Q) −→ Q×p be a character of Gal(K/Q), not necessarily faithful. We study the Fittingideal of the ψ-quotient (Sel(OK∞

[1/m],E[p∞])∨)ψ = Sel(OK∞[1/m],E[p∞])∨⊗Zp[Gal(K/Q)] Oψ . We denote

by F the subfield of K corresponding to the kernel of ψ . We regard ψ as a faithful character of Gal(F/Q).Since Sel(OK∞

[1/m],E[p∞])Gal(K/F) = Sel(OF∞[1/m],E[p∞]), we have

(Sel(OK∞[1/m],E[p∞])∨)ψ = (Sel(OF∞

[1/m],E[p∞])∨)ψ

where the right hand side is defined to be Sel(OF∞[1/m],E[p∞])∨⊗Zp[Gal(F/Q)] Oψ .

We put Λψ = (ΛF∞)ψ . The group homomorphism ψ induces the ring homomorphism ΛF∞

−→ Λψ

which we also denote by ψ . The composition with cK∞/F∞: ΛK∞

−→ ΛF∞and the above ring homo-

morphism ψ is also denoted by ψ : ΛK∞−→ Λψ . Note that F/Q is a cyclic extension of degree a

power of p. We denote by F ′ the subfield of F such that [F : F ′] = p. We put N0 = NGal(F/F ′) =

Σσ∈Gal(F/F ′)σ . If we put [F : Q] = pc and take a generator γ of Gal(F/Q), N0 = Σp−1i=0 γ pc−1i is a cy-

clotomic polynomial and Oψ = Zp[µpc ] ' Zp[Gal(F/Q)]/N0. For any Zp[Gal(F/Q)]-module M, wedefine Mψ = Ker(N0 : M −→ M). Then the Pontrjagin dual of Mψ is (Mψ)∨ = (M∨)/N0 = (M∨)ψ .By the same method as the proof of (6), we have H1(Gal(F/F ′),Sel(OF∞

[1/m],E[p∞])) = 0. There-fore, σ − 1 : Sel(OF∞

[1/m],E[p∞]) −→ Sel(OF∞[1/m],E[p∞])ψ is surjective where σ = γ pc−1

is a gener-ator of Gal(F/F ′). Therefore, taking the dual, we know that there is an injective homomorphism from(Sel(OF∞

[1/m],E[p∞])∨)ψ to Sel(OF∞[1/m],E[p∞])∨ which is a free Zp-module. Therefore, (Sel(OF∞

[1/m],E[p∞])∨)ψ

contains no nontrivial finite Λψ -submodule. This shows that

Fitt0,Λψ((Sel(OF∞

[1/m],E[p∞])∨)ψ) = char((Sel(OF∞[1/m],E[p∞])∨)ψ).

Consider the ψ-quotient of the exact sequence (4);

(⊕v|m

Tp(E)Γv)ψ −→ (Sel(OF∞[1/m],E[p∞])∨)ψ −→ (Sel(OF∞

,E[p∞])∨)ψ −→ 0

where v runs over all primes of F∞ above m. Since Ext1Zp[Gal(F/Q)](Oψ ,Sel(OF∞,E[p∞]))= H0(Gal(F/Q),Sel(OF∞

,E[p∞]))

is finite, the first map of the above exact sequence has finite kernel.


Suppose that ` is a prime divisor of m. If ` is unramified in F , we have

Fitt0,Λψ((⊕v|`

Tp(E)Γv)ψ) = P′`(σ`)Λψ

where P′`(x) = x2−a`x+`. If ` is ramified in F , ψ(`) = 0 and (⊕

v|` Tp(E)Γv)ψ is finite. Therefore, we have

char((⊕v|m

Tp(E)Γv)ψ) = ( ∏`∈S\Sram(F)

P′`(σ`))Λψ .

Using the above exact sequence and Kato’s theorem ψ(ξF∞) ∈ char((Sel(OF∞

,E[p∞])∨)ψ), we have

char((Sel(OF∞[1/m],E[p∞])∨)ψ)⊃ ψ(ξF∞

)( ∏`∈S\Sram(F)

P′`(σ`))Λψ .

Since ξF∞(∏`∈S\Sram(F) P′`(σ`)) = ξF∞,S modulo unit and cK∞/F∞

(ξK∞,S) = ξF∞,S by (3), we obtain

ψ(ξK∞,S) ∈ Fitt0,(ΛK∞)ψ((Sel(OK∞

[1/m],E[p∞])∨)ψ) (7)

for any character ψ of Gal(K/Q). Since the µ-invariant of Sel(OK∞[1/m],E[p∞])∨ is zero as we explained

above, (7) impliesξK∞,S ∈ Fitt0,ΛK∞

(Sel(OK∞[1/m],E[p∞])∨)

(see Lemma 4.1 in [9], for example).

(2) We use the same notation ψ , F , etc. as above. At first, we assume (1). Then the algebraic λ -invariantof Sel(E/F∞,E[p∞])∨ equals the analytic λ -invariant by Hachimori and Matsuno [6], [13], so the mainconjecture char((Sel(OF∞

,E[p∞])∨)ψ) = ψ(ξF∞)Λψ also holds. Therefore, we have

char((Sel(OF∞[1/m],E[p∞])∨)ψ) = ψ(ξF∞

)( ∏`∈S\Sram(F)

P′`(σ`))Λψ

= ψ(ξF∞,S)Λψ = ψ(ξK∞,S)Λψ

andFitt0,(ΛK∞

)ψ((Sel(OK∞

[1/m],E[p∞])∨)ψ) = ψ(ξK∞,SΛK∞)Λψ .

It follows from [9] Corollary 4.2 that

Fitt0,ΛK∞(Sel(OK∞

[1/m],E[p∞])∨) = ξK∞,SΛK∞.

On the other hand, if we assume the above equality, taking the Gal(K/Q)-invariant part of Sel(OK∞[1/m],E[p∞]),

we getFitt0,ΛQ∞

(Sel(OQ∞[1/m],E[p∞])∨) = ξQ∞,SΛQ∞,S,

which implies (1).

2.3 An analogue of Stickelberger’s theorem

Let K/Q be a finite abelian p-extension. When the conductor of K is m, we define ϑK ∈RK =Zp[Gal(K/Q)]to be the image of ϑQ(µmp∞ ) ∈ ΛQ(µmp∞ ). Therefore, if m is prime to p, ϑK is the image of ϑQ(µm) = (1−σpα)(1− σ−1

pα

)θQ(µm) by (4). If m = m′pn for some m′ which is prime to p and for some n ≥ 2, ϑK is theimage of ϑQ(µm′ pn ) = α−n(θQ(µm′ pn )−α−1νm′pn,m′pn−1(θQ(µm′ pn−1 ))).

For any positive integer n, we denote by Q(n) the maximal p-subextension of Q in Q(µn).

Theorem 2.3.1 For any finite abelian p-extension K in which all bad primes of E are unramified, ϑKannihilates Sel(OK ,E[p∞])∨, namely we have

ϑK Sel(OK ,E[p∞])∨ = 0.

236 Masato Kurihara

Proof. We may assume K = Q(mpn) for some squarefree product m of primes in Pgood and for somen ∈ Z≥0. By Theorem 2.3.1 (1), taking S to be the set of all prime divisors of m, we have ξK∞

∈Fitt0,ΛK∞

(Sel(OK∞[1/m],E[p∞])∨), which implies ξK∞

Sel(OK∞,E[p∞])∨ = 0. Let ξK ∈ RK = Zp[Gal(K/Q)]

be the image of ξK∞. Since the natural map Sel(OK ,E[p∞]) −→ Sel(OK∞

,E[p∞]) is injective, we haveξK Sel(OK ,E[p∞])∨ = 0.

By the definitions of ξQ(µmp∞ ), ξQ(mpn), ϑQ(mpn), we can write

ξK = ξQ(mpn) = ϑQ(mpn)+ ∑d|m,d 6=m

λdνm,d(ϑQ(d pn)) (1)

for some λd ∈ RQ(mpn) where νm,d : RQ(d pn) −→ RQ(mpn) is the norm map defined similarly as in §2.1. Wewill prove this theorem by induction on m. Since d <m, we have ϑQ(d pn) ∈AnnRQ(d pn)(Sel(OQ(d pn),E[p∞])∨)

by the hypothesis of the induction. This implies that νm,d(ϑQ(d pn)) annihilates Sel(OQ(mpn),E[p∞])∨. SinceξK is in AnnRK (Sel(OK ,E[p∞])∨), the above equation implies that ϑK is in AnnRK (Sel(OK ,E[p∞])∨).

Remark 2.3.2 Let K, S, m be as in Theorem 2.2.2. Under our assumptions, the control theorem workscompletely;

Sel(OK [1/m],E[p∞])'−→ Sel(OK∞

[1/m],E[p∞])Gal(K∞/K).

Therefore, Theorem 2.2.2 (1) implies that Fitt0,RK (Sel(OK [1/m],E[p∞])∨) is principal and

ξK,S ∈ Fitt0,RK (Sel(OK [1/m],E[p∞])∨) (2)

where ξK,S is the image of ξK∞,S in RK .Theorem 2.2.2 (2) implies that if we assume the main conjecture (1), we have

Fitt0,RK (Sel(OK [1/m],E[p∞])∨) = ξK,SRK . (3)

2.4 Higher Fitting ideals

For a commutative ring R and a finitely presented R-module M with n generators, let A be an n×m relationmatrix of M. For an integer i≥ 0, Fitti,R(M) is defined to be the ideal of R generated by all (n− i)× (n− i)minors of A (cf. [19]; this ideal Fitti,R(M) does not depend on the choice of a relation matrix A).

Suppose that K/Q is a finite extension such that K is in the cyclotomic Zp-extension Q∞ of Q, and thatm is a squarefree product of primes in P(N). We define K(m) by K(m) = Q(m)K.

We put G` = Gal(Q(`)/Q) and Gm = Gal(Q(m)/Q) = Π`|mG`. We have Gal(K(m)/K) = Gm. We putn` = ordp(`− 1). Suppose that m = `1 · ... · `r. We take a generator τì of Gì and put Si = τì − 1 ∈ RK(m).We write ni for nì . We identify RK(m) with

RK [Gm] = RK [S1, ...,Sr]/((1+S1)pn1 −1, ...,(1+Sr)

pnr −1).

We consider ϑK(m) ∈ RK(m) and write

ϑK(m) = ∑i1,...,ir≥0

a(m)i1,...,ir

Sii1 · ... ·S

irr

where a(m)i1,...,ir

∈ RK . Put n0 = minn1, ...,nr. For s ∈ Z>0, we define cs to be the maximal positive integerc such that

T−1((1+T )pn0 −1) ∈ pcZp[T ]+T s+1Zp[T ].

For example, c1 = n0, ..., cp−2 = n0, cp−1 = n0− 1, ..., cp2−1 = n0− 2. If i1,...,ir ≤ s, a(m)i1,...,ir

mod pcs is

well-defined (it does not depend on the choice of a(m)i1,...,ir

).

Theorem 2.4.1 Let K be an intermediate field of the cyclotomic Zp-extension Q∞/Q with [K : Q]< ∞. Letcs be the integer defined above for s ∈Z>0 and m. Assume that i1,...,ir ≤ s and i1+ ...+ ir ≤ i. Then we have

a(m)i1,...,ir

∈ Fitti,RK/pcs (Sel(E/K,E[pcs ])∨).


For m= `1 · ... ·`r, we denote (−1)r times the coefficient of S1 · ... ·Sr in ϑK(m) by δm. If ì splits completelyin K for all i = 1,...,r, we can write

ϑK(m) ≡ δm

r

∏i=1

(1− τì) = (−1)rδmS1 · ... ·Sr (mod pN ,S2

1, ...,S2r ) (1)

(see [12] §6.3). Taking s = 1 and i = r in Theorem 2.4.1, we get

Corollary 2.4.2 Let K/Q be a finite extension such that K ⊂Q∞. We have

δm ∈ Fittr,RK/pN (Sel(E/K,E[pN ])∨)

where m = `1 · ... · `r.

Proof of Theorem 2.4.1. We may assume K = Q(pn) for some n ≥ 0, so K(m) = Q(mpn). First of all,we consider the image ξK(m) ∈ RK(m) of ξK(m)∞

. Since Sel(E/K(m),E[p∞]) −→ Sel(E/K(m)∞,E[p∞]) isinjective, ξK(m) is in Fitt0,RK(m)

(Sel(E/K(m),E[p∞])∨) by Theorem 2.2.2 (1). We write

ξK(m) = ∑i1,...,ir≥0

α(m)i1,...,ir

Sii1 · ... ·S

irr

where α(m)i1,...,ir

∈ RK . Assume that i1,...,ir ≤ s and i1 + ...+ ir ≤ i. Then by Lemma 3.1.1 in [12] we have

α(m)i1,...,ir

∈ Fitti,RK/pcs (Sel(E/K,E[pcs ])∨).

On the other hand, since K(m) = Q(mpn) for some n≥ 0, we have

ξK(m) = ϑK(m)+ ∑d|m,d 6=m

λdνm,d(ϑQ(d pn))

for some λd ∈ RK(m) by (1). This implies that the images of ξK(m) and ϑK(m) under the canonical homomor-phism

RK(m) = RK [S1, ...,Sr]/I −→ RK [[S1, ...,Sr]]/J

coincide where I = ((1+ S1)pn1 − 1, ...,(1+ S1)

pnr − 1) and J = (S−11 (1+ S1)

pn1 − 1, ...,S−1r (1+ S1)

pnr −1,Ss+1

1 , ...,Ss+1r ). Therefore, α

(m)i1,...,ir

≡ a(m)i1,...,ir

mod pcs for i1,...,ir ≤ s. It follows that a(m)i1,...,ir

∈Fitti,RK/pcs (Sel(E/K,E[pcs ])∨).This completes the proof of Theorem 2.4.1.

3 Review of Kolyvagin systems of Gauss sum type for elliptic curves

In this section, we recall the results in [12] on Euler systems and Kolyvagin systems of Gauss sum type inthe case of elliptic curves. From this section we assume all the assumptions (i), (ii), (iii), (iv) in §1.1.

3.1 Some definitions

Recall that in §2 we defined Pgood by Pgood = ` | ` is a good reduction prime for E \p, and P(N) by

P(N) = ` ∈ Pgood | `≡ 1 (mod pN)

for a positive integer N > 0. If ` is in Pgood , the absolute Galois group GFàcts on the group E[pN ] of

pN-torsion points, so we consider H i(F`,E[pN ]). We define

P(N)0 = ` ∈ P(N) | H0(F`,E[pN ]) contains an element of order pN,

(P′0)(N) = ` ∈ P(N) | H0(F`,E[pN ]) = E[pN ], and

P(N)1 = ` ∈ P(N) | H0(F`,E[pN ])' Z/pN.

238 Masato Kurihara

So P(N)0 ⊃ (P′0)

(N), P(N)0 ⊃ P

(N)1 , and (P′0)

(N)∩P(N)1 = /0. Suppose that ` is in P

(N)1 . Then, since `≡ 1 (mod

pN), we have an exact sequence 0 −→ Z/pN −→ E[pN ] −→ Z/pN −→ 0 of GF`-modules where GF`

acts

on Z/pN trivially. So the action of the Frobenius Frob` at ` on E[pN ] can be written as(

1 10 1

)for a suitable

basis of E[pN ]. Therefore, H1(F`,E[pN ]) is also isomorphic to Z/pN for ` ∈ P(N)1 .

Let t ∈ E[pN ] be an element of order pN . We define

P(N)0,t = ` ∈ P(N) | t ∈ H0(F`,E[pN ]),

P(N)1,t = ` ∈ P(N) | H0(F`,E[pN ]) = (Z/pN)t.

So, P(N)0 =

⋃t P

(N)0,t and P

(N)1 =

⋃t P

(N)1,t where t runs over all elements of order pN . Since we assumed that

the Galois action on the Tate module is surjective, both (P′0)(N) and P

(N)1,t are infinite by Chebotarev density

theorem ([12] §4.3).We define K(p) to be the set of number fields K such that K/Q is a finite abelian p-extension in which

all bad primes of E are unramified. Suppose that K is in K(p). We define

(P′0)(N)(K) = ` ∈ (P′0)

(N) | ` splits completely in K,

P(N)1 (K) = ` ∈ P

(N)1 | ` splits completely in K.

Again by Chebotarev density theorem, both (P′0)(N)(K) and P

(N)1 (K) are infinite ([12] §4.3).

Suppose `∈Pgood . For a prime v above `, we know H1(Kv,E[pN ])/E(Kv)⊗Z/pN =H0(κ(v),E[pN ](−1))where κ(v) is the residue field of v. We put

H2` (K) =

⊕v|`

H0(κ(v),E[pN ](−1)). (1)

If ` is in (P′0)(N)(K) (resp. P(N)

1 (K)), H2` (K) is a free RK/pN-module of rank 2 (resp. rank 1) where RK =

Zp[Gal(K/Q)] as before.

From now on, for a prime `∈P(N)0 , we fix a prime `Q of an algebraic closure Q above `. For any algebraic

number field F , we denote the prime of F below `Q by `F , so when we consider finite extensions F1/k, F2/ksuch that F1 ⊂ F2, the primes `F2 , `F1 satisfy `F2 |`F1 .

We take a primitive pn-th root of unity ζpn such that (ζpn)n≥1 ∈ Zp(1) = lim←−

µpn , and fix it.

In the following, for each ` in P(N)0 (K), we take t` ∈ H0(F`,E[pN ]) and fix it. We define

t`,K = (t`⊗ζ⊗(−1)pN ,0, ...,0) ∈H2

` (K) (2)

where the right hand side is the element whose `K-component is t`⊗ζ⊗(−1)pN and other components are zero.

Suppose that K is in K(p). Let K∞/K be the cyclotomic Zp-extension, and Kn be the n-th layer.Since Sel(OK∞

,E[p∞])∨ is a finitely generated Zp-module, the corestriction map Sel(OKm ,E[pN ]) −→

Sel(OK ,E[pN ]) is the zero map if m is sufficiently large. We take the minimal m > 0 satisfying this property,and put K[1] = Km. We define inductively K[n] by K[n] = (K[n−1])[1] where we applied the above definition toK[n−1] instead of K.

We can compute how large K[n] is. Let λ be the λ -invariant of Sel(OK∞,E[p∞])∨. We take a ∈ Z≥0

such that pa+1 − pa ≥ λ . Suppose that K = K′m (m-th layer of K′∞/K′) for some K′ such that p is un-ramified in K′. The corestriction map Sel(OK′a+1

,E[p]) −→ Sel(OK′a ,E[p]) is the zero map. Therefore,Sel(OK′a+N

,E[pN ])−→Sel(OK′a ,E[pN ]) is the zero map. Put a′=max(a−m,0). Then Sel(OKa′+N

,E[pN ])−→Sel(OKa′ ,E[p

N ]) is the zero map. Therefore, we have K[1] ⊂ Ka′+N . Also we know K[n] ⊂ Ka′+nN .

Let nλ , dn be the numbers defined just before (1) in §1.2. Then we can show that if ` ∈ P(N)1 satisfies

`≡ 1 (mod pdn ), ` is in P(N)1 (Q[n]) by the same method as above.


3.2 Euler systems of Gauss sum type for elliptic curves

We use the following lemma which is the global duality theorem (see Theorem 2.3.4 in Mazur and Rubin[14]).

Lemma 3.2.1 Suppose that m is a product of primes in Pgood . We have an exact sequence

0−→ Sel(OK ,E[pN ])−→ Sel(OK [1/m],E[pN ])−→⊕`|m

H2` (K)−→ Sel(OK ,E[pN ])∨.

We remark that we can take m such that the last map is surjective in our case (see Lemma 3.4.1 below).

Let K be a number field in K(p) and ` ∈ P(N)0 (K[1]). We apply the above lemma to K[1] and obtain an

exact sequence

Sel(OK[1] [1/`],E[pN ])

∂`−→H2` (K[1])

w`−→ Sel(OK[1] ,E[pN ])∨.

Consider ϑK[1]t`,K[1] ∈H2` (K[1]). By Theorem 2.3.1 we know w`(ϑK[1]t`,K[1]) = ϑK[1]w`(t`,K[1]) = 0. Therefore,

there is an element g ∈ Sel(OK[1] [1/`],E[pN ]) such that ∂`(g) = ϑK[1]t`,K[1] . We define

g(K)`,t`

= CorK[1]/K(g) ∈ Sel(OK [1/`],E[pN ]). (1)

This element g(K)`,t`

does not depend on the choice of g ∈ Sel(OK[1] [1/`],E[pN ]) ([12] §5.4). We write g`

instead of g(K)`,t`

when no confusion arises.

Remark 3.2.2 To define g`, we used in [12] the p-adic L-function θK∞whose Euler factor at ` is 1−

a`` σ−1` + 1

`σ−2` . The element θK∞

can be constructed from ϑK∞by the same method as when we constructed

ξK∞in §2.1. In the above definition (1), we used ϑK (namely ϑK∞

) instead of θK∞.

3.3 Kolyvagin derivatives of Gauss sum type

Let ` be a prime in Pgood . We define ∂` as a natural homomorphism

∂` : H1(K,E[pN ])−→H2` (K) =

⊕v|`

H0(κ(v),E[pN ](−1))

where we used H1(Kv,E[pN ])/E(Kv)⊗Z/pN = H0(κ(v),E[pN ](−1)).Next, we assume ` ∈ P

(N)1 (K). We denote by Q`(`) the maximal p-subextension of Q` inside Q`(µ`).

Put G` = Gal(Q`(`)/Q`). By Kummer theory, G` is isomorphic to µpn` where n` = ordp(`−1). We denoteby τ` the corresponding element of G` to ζpn` that is the primitive pn`-th root of unity we fixed.

We consider the natural homomorphism H1(Q`,E[pN ]) −→ H1(Q`(`),E[pN ]) and denote the kernelby H1

tr(Q`,E[pN ]). Let Q`,nr be the maximal unramified extension of Q`. We identify H1(F`,E[pN ]) withH1(Gal(Q`,nr/Q`),E[pN ]), and regard it as a subgroup of H1(Q`,E[pN ]). Then both H1(F`,E[pN ]) andH1

tr(Q`,E[pN ]) are isomorphic to Z/pN , and we have decomposition

H1(Q`,E[pN ]) = H1(F`,E[pN ])⊕H1tr(Q`,E[pN ])

as an abelian group. We also note that H1(F`,E[pN ]) coincides with the image of the Kummer map and isisomorphic to E(Q`)⊗Z/pN . We consider the homomorphism

φ′ : H1(Q`,E[pN ])−→ H1(F`,E[pN ]) (1)

which is obtained from the above decomposition.Note that H1(F`,E[pN ]) = E[pN ]/(Frob`−1) where Frob` is the Frobenius at `. Since ` is in P

(N)1 ,

Frob−1` −1 : E[pN ]/(Frob`−1) −→ E[pN ]Frob`=1 = H0(F`,E[pN ]) is an isomorphism. We define φ ′′ :

H1(Q`,E[pN ])−→ H0(F`,E[pN ]) as the composition of φ ′ and H1(F`,E[pN ])Frob−1

` −1−→ H0(F`,E[pN ]). We

define

240 Masato Kurihara

φ` : H1(K,E[pN ])−→H2` (K)(1)

as the composition of the natural homomorphism H1(K,E[pN ]) −→⊕

v|` H1(Kv,E[pN ]) and φ ′′ for Kv.Using the primitive pN-th root of unity ζpN we fixed, we regard φ` as a homomorphism

φ` : H1(K,E[pN ])−→H2` (K).

For a prime ` ∈ P(N)1 (K), we put G` = Gal(Q(`)/Q). We identify G` with Gal(Q`(`)/Q`). Recall that we

defined n` by pn` = [Q(`) : Q], and we took a generator τ` of G` above. We define

N` =pn`−1

∑i=0

τi` ∈ Z[G`], D` =

pn`−1

∑i=0

iτ i` ∈ Z[G`]

as usual.We define N

(N)1 (K) to be the set of squarefree products of primes in P

(N)1 (K). We suppose 1 ∈N

(N)1 (K).

For m ∈N(N)1 (K), we put Gm = Gal(Q(m)/Q), Nm = Π`|mN` ∈ Z[Gm], and Dm = Π`|mD` ∈ Z[Gm]. Assume

that ` is in (P′0)(N)(K(m)[1]) and consider gK(m)

`,t`∈ Sel(OK(m)[1/`],E[pN ]). We can check that DmgK(m)

`,tìs

in Sel(OK(m)[1/m`],E[pN ])Gm . Using the fact that Sel(OK [1/m`],E[pN ])'−→ Sel(OK(m)[1/m`],E[pN ])Gm is

bijective by Lemma 3.3.1 below (cf. also [12] Lemma 6.3.1), we define

κm,` = κ(K)m,`,t`

∈ Sel(OK [1/m`],E[pN ]) (2)

to be the unique element whose image in Sel(OK(m)[1/m`],E[pN ]) is Dmg(K(m))`,t`

.The following lemma will be also used in the next section.

Lemma 3.3.1 Suppose that K, L∈K(p) and K⊂L. For any m∈Z>0, the restriction map Sel(OK [1/m],E[pN ])'−→

Sel(OL[1/m],E[pN ])Gal(L/K) is bijective.

Proof. Let NE be the conductor of E, m′ = mpNE , and m′′ the product of primes which divide pNE andwhich do not divide m. Put G = Gal(L/K). We have a commutative diagram of exact sequences

0 −→ Sel(OK [1/m],E[pN ]) −→ Sel(OK [1/m′],E[pN ]) −→⊕

v|m′′H2K,vyα1

yα2

yα3

0 −→ Sel(OL[1/m],E[pN ])G −→ Sel(OL[1/m′],E[pN ])G −→ (⊕

w|m′′H2L,w)

G

where H2K,v =H1(Kv,E[pN ])/E(Kv)⊗Z/pN and H2

L,w =H1(Lw,E[pN ])/E(Lw)⊗Z/pN . Since Sel(OL[1/m′],E[pN ])=

H1et(SpecOL[1/m′],E[pN ]) and H0(L,E[pN ]) = 0, α2 is bijective. Suppose that v divides m′′ and w is above

v. When v divides NE , since v is unramified in L and p is prime to Tam(E), H2K,v −→ H2

L,w is injective(Greenberg [3] §3). When v is above p, H2

K,v −→ H2L,w is injective because ap 6≡ 1 (mod p) (Greenberg [3]

§3). Hence α3 is injective. Therefore, α1 is bijective.

In [11], if m has a factorization m = `1 · ... · `r such that ì+1 ∈ P(N)1 (K(`1 · ... · ì)) for all i = 1,...,r− 1,

we called m well-ordered. But the word “well-ordered" might cause confusion, so we call m admissible inthis paper if m satisfies the above condition. Note that we do not impose the condition `1 < ... < `r in theabove definition, and that m is admissible if there is one factorization as above. We sometimes call the setof prime divisors of m admissible if m is admissible.

Suppose that m = `1 · ... · `r. We define δm ∈ RK/pN by

ϑK(m) ≡ δm

r

∏i=1

(1− τì) (mod pN ,(τ`1 −1)2, ...,(τ`r −1)2) (3)

(see [12] §6.3).We simply write κm,` for κ

(K)m,`,t`

. We have the following Proposition ([12] Propositions 6.3.2, 6.4.5 andLemma 6.3.4).

Proposition 3.3.2 Suppose that m is in N(N)1 (K), and ` ∈ (P′0)

(N)(K(m)[1]). We take n0 sufficiently largesuch that every prime of Kn0 dividing m is inert in K∞/Kn0 . We further assume that ` ∈ (P′0)

(N)(Kn0+N).


Then(0) κm,` ∈ Sel(OK [1/m`],E[pN ]).(1) ∂r(κm,`) = φr(κ m

r ,`) for any prime divisor r of m.

(2) ∂`(κm,`) = δmt`,K .(3) Assume further that m is admissible. Then φr(κm,`) = 0 for any prime divisor r of m.

3.4 Construction of Kolyvagin systems of Gauss sum type

In the previous subsection we constructed κm,` for m∈N(N)1 (K) and a prime `∈ (P′0)(N)(K) satisfying some

properties. In this subsection we construct κm,` for ` ∈ P(N)1 (K) satisfying some properties (see Proposition

3.4.2). The property (4) in Proposition 3.4.2 is a beautiful property of our Kolyvagin systems of Gauss sumtype, which is unique for Kolyvagin systems of Gauss sum type.

For a squarefree product m of primes, we define ε(m) to be the number of prime divisors of m, namelyε(m) = r if m = `1 · ... · `r.

For any prime number `, we write H2` (K) =

⊕v|` H1(Kv,E[pN ])/E(Kv)⊗Z/pN , and consider the natural

mapwK :

⊕`

H2` (K)−→ Sel(OK ,E[pN ])∨

which is obtained by taking the dual of Sel(OK ,E[pN ])−→⊕

v E(Kv)⊗Z/pN . We also consider the naturalmap

∂K : H1(K,E[pN ])−→⊕`

H2` (K).

We use the following lemma which was proved in [12] Proposition 4.4.3 and Lemma 6.2.1 (2).

Lemma 3.4.1 Suppose that K ∈ K(p) and r1,...,rs are s distinct primes in P(N)1 (K). Assume that for each

i = 1,...,s, σi ∈H2ri(K) is given, and also x ∈ Sel(OK ,E[pN ])∨ is given. Let K′/K be an extension such that

K′ ∈K(p). Then there are infinitely many ` ∈ P(N)0 (K) such that wK(t`,K) = x. We take such a prime ` and

fix it. Then there are infinitely many `′ ∈ (P′0)(N)(K′) which satisfy the following properties:

(i) wK(t`′,K) = wK(t`,K) = x.(ii) There is an element z ∈ Sel(OK [1/``′],E[pN ]) such that ∂K(z) = t`′,K − t`,K and φri(z) = σi for eachi = 1,...,s.

Assume that m` is in N(N)1 (K[ε(m`)]). By Lemma 3.4.1 we can take `′ ∈ (P′0)

(N) satisfying the followingproperties:(i) `′ ∈ (P′0)

(N)(K[ε(m`)](m)[1]Kn0+N) where n0 is as in Proposition 3.3.2.(ii) wK[ε(m`)]

(t`′,K[ε(m`)]) = wK[ε(m`)]

(t`,K[ε(m`)]).

(iii) Let φ(K[ε(m`)])r : H1(K[ε(m`)],E[pN ])−→H2

r (K[ε(m`)]) be the map φr for K[ε(m`)]. There is an element b′ inSel(OK[ε(m`)]

[1/``′],E[pN ]) such that

∂K[ε(m`)](b′) = t`′,K[ε(m`)]

− t`,K[ε(m`)]

and φK[ε(m`)]r (b′) = 0 for all r dividing m.

We have already defined κm,`′ in the previous subsection. We put b = CorK[ε(m`)]/K(b′) and define

κm,` = κm,`′ −δmb. (1)

Then this element does not depend on the choice of `′ and b′ (see [12] §6.4). In [12], we took b′ which doesnot necessarily satisfy φ

K[ε(m`)]r (b′) = 0 in the definition of κm,`. But we adopted the above definition here

because it is simpler and there is no loss of generality.The next proposition was proved in [12] Propositions 6.4.3, 6.4.5, 6.4.6.

Proposition 3.4.2 Suppose that m` is in N(N)1 (K[ε(m`)]). Then

(0) κm,` ∈ Sel(OK [1/m`],E[pN ]).(1) ∂r(κm,`) = φr(κ m

r ,`) for any prime divisor r of m.

(2) ∂`(κm,`) = δmt`,K .

242 Masato Kurihara

(3) Assume further that m is admissible. Then φr(κm,`) = 0 for any prime divisor r of m.

(4) Assume further that m` is admissible, and m` is in N(N)1 (K[ε(m`)+1]). Then we have

φ`(κm,`) =−δm`t`,K .

4 Relations of Selmer groups

In this section, we prove a generalized version of Theorem 1.2.3.

4.1 Injectivity theorem

Suppose that K is in K(p) and that m is in N(N)1 (K). For a prime divisor r of m, we denote by

wr : H2r (K)−→ Sel(OK ,E[pN ])∨

the homomorphism which is the dual of Sel(OK ,E[pN ]) −→⊕

v|r E(Kv)⊗Z/pN . Recall that H2r (K) is a

free RK/pN-module of rank 1, generated by tr,K .

Proposition 4.1.1 We assume that δm is a unit of RK/pN for some m ∈ N(N)1 (K). Then the natural homo-

morphism ⊕r|mwr :⊕

r|mH2r (K)−→ Sel(OK ,E[pN ])∨ is surjective.

Remark 4.1.2 We note that δm is numerically computable, in principle.

Proof of Proposition 4.1.1. Let x be an arbitrary element in Sel(OK ,E[pN ])∨. Let wr :H2r (K)−→ Sel(OK ,E[pN ])∨

be the natural homomorphism for each r | m. We will prove that x is in the submodule generated by allwr(tr,K) for r |m. Using Lemma 3.4.1, we can take a prime `∈ (P′0)(N)(K(m)[1]Kn0+N) such that w`(t`,K) = xand ` is prime to m. We consider the Kolyvagin derivative κm,` which was defined in (2). Consider the exactsequence

Sel(OK [1/m`],E[pN ])∂−→⊕`′|m`

H2`′(K)

wK−→ Sel(OK ,E[pN ])∨

(see Lemma 3.2.1) where ∂ = (⊕∂`′)`′|m` and wK((z`′)`′|m`) = ∑`′|m` w`′(z`′). For each r | m we defineλr ∈ RK/pN by ∂r(κm,`) = λrtr,K ∈H2

r (K). The above exact sequence and Proposition 3.3.2 (2) imply that

δmx+∑r|m

λrwr(tr,K) = 0

in Sel(OK ,E[pN ])∨. Since we assumed that δm is a unit, x is in the submodule generated by all wr(tr,K)’s.This completes the proof of Proposition 4.1.1.

For a prime ` ∈ P(N)1 (K), we define

H1`, f (K) =

⊕v|`

E(κ(v))⊗Z/pN .

Since κ(v) = F`, E(κ(v))⊗Z/pN is isomorphic to Z/pN and H1`, f (K) is a free RK/pN-module of rank 1.

Corollary 4.1.3 Suppose that m = `1 · ... · `a is in N(N)1 (K). We assume that δm is a unit of RK/pN . Then

the natural homomorphism

sm : Sel(OK ,E[pN ])−→a⊕

i=1

H1`i, f (K)

is injective.

Proof. This is obtained by taking the dual of the statement in Proposition 4.1.1.


4.2 Relation matrices

Theorem 4.2.1 Suppose that m = `1 · ... ·à is in N(N)1 (K[a+1]). We assume that m is admissible and that δm

is a unit of RK/pN . Then(1) Sel(OK [1/m],E[pN ]) is a free RK/pN-module of rank a.(2) κ m

ì,ì1≤i≤a is a basis of Sel(OK [1/m],E[pN ]).

(3) The matrix

A=

δ m

`1φ`1(κ m

`1`2,`2) ... φ`1(κ m

`1à,à)

φ`2(κ m`1`2

,`1) δ m`2

... φ`2(κ m`2à

,à)

. .

. .φà(κ m

`1à,`1) φà(κ m

`2à,`2) ... δ m

à

(1)

is a relation matrix of Sel(E/K,E[pN ])∨.

In particular, if a = 2, the above matrix is A =

(δ`2 φ`1(g`2)

φ`2(g`1) δ`1

). This is described in Remark 10.6

in [11] in the case of ideal class groups.

Proof of Theorem 4.2.1 (1). By Proposition 4.1.1,⊕a

i=1H2ì(K) −→ Sel(OK ,E[pN ])∨ is surjective. There-

fore, by Lemma 3.2.1 we have an exact sequence

0−→ Sel(OK ,E[pN ]) −→ Sel(OK [1/m],E[pN ])∂−→

a⊕i=1

H2ì(K)

−→ Sel(OK ,E[pN ])∨ −→ 0. (2)

It follows that #Sel(OK [1/m],E[pN ]) = #⊕a

i=1H2ì(K) = #(RK/pN)a.

Let mRK be the maximal ideal of RK . By Lemma 3.3.1, Sel(Z[1/m],E[pN ])'−→Sel(OK [1/m],E[pN ])Gal(K/Q)

is bijective. Since H0(Q,E[p∞]) = 0, the kernel of the multiplication by p on Sel(Z[1/m],E[pN ]) isSel(Z[1/m],E[p]). Therefore, we have an isomorphism Sel(OK [1/m],E[pN ])∨⊗RK RK/mRK 'Sel(Z[1/m],E[p])∨.From the exact sequence

0−→ Sel(Z,E[p])−→ Sel(Z[1/m],E[p])−→a⊕

i=1

H2ì(Q)−→ Sel(Z,E[p])∨ −→ 0,

and H2ì(Q) = H0(Fì ,E[p])' Fp, we know that Sel(Z[1/m],E[p]) is generated by a elements. Therefore,

by Nakayama’s lemma, Sel(OK [1/m],E[pN ])∨ is generated by a elements. Since #Sel(OK [1/m],E[pN ])∨ =#(RK/pN)a, Sel(OK [1/m],E[pN ])∨ is a free RK/pN-module of rank a. This shows that Sel(OK [1/m],E[pN ])is also a free RK/pN-module of rank a because RK/pN is a Gorenstein ring.

(2) We identify⊕a

i=1H2ì(K) with (RK/pN)a, using a basis tì,K1≤i≤a. Consider φì : Sel(OK [1/m],E[pN ])−→

H2ì(K) and the direct sum of φì , which we denote by Φ ;

Φ =⊕ai=1φì : Sel(OK [1/m],E[pN ])−→

a⊕i=1

H2ì(K)' (RK/pN)a.

Recall that κ mì,ì is an element of Sel(OK [1/m],E[pN ]) (Proposition 3.4.2 (0)). By Proposition 3.4.2 (3), (4),

we haveΦ(κ m

ì,ì) =−δmei

for each i where ei1≤i≤a is the standard basis of the free module (RK/pN)a. Since we are assuming thatδm is a unit, Φ is surjective. Since both the target and the source are free modules of the same rank, Φ isbijective. This implies Theorem 4.2.1 (2).

(3) Using the exact sequence (2) and the isomorphism Φ , we have an exact sequence

244 Masato Kurihara

(Rn/pN)a ∂Φ−1−→

⊕1≤i≤a

H2ì(Kn)

r−→ Sel(OK ,E[pN ])∨ −→ 0.

We take a basis −δmei1≤i≤a of (Rn/pN)a and a basis tì,K1≤i≤a of⊕

1≤i≤aH2ì(Kn). Then the (i, j)-

component of the matrix corresponding to ∂ Φ−1 is ∂ì(κ m` j,` j). If i = j, this is δ m

ìby Proposition 3.4.2 (2).

If i 6= j, we have ∂ì(κ m` j,` j) = φì(κ m

ì` j,` j) by Proposition 3.4.2 (1). This completes the proof of Theorem

4.2.1.

Remark 4.2.2 Suppose that ` is in P(N)1 (K). We define

Φ′` : H1(K,E[pN ])−→H1

`, f (K)

as the composition of the natural map H1(K,E[pN ]) −→⊕

v|` H1(Kv,E[pN ]) and φ ′ : H1(Kv,E[pN ]) −→H1(κ(v),E[pN ]) = E(κ(v))⊗Z/pN in (1). For m ∈N

(N)1 (K), we define

Φ′m : H1(K,E[pN ])−→

⊕`|m

H1`, f (K)

as the direct sum of Φ ′` for ` |m. By definition, the restriction of Φ ′m to S= Sel(E/K,E[pN ]) coincides withthe canonical map sm;

(Φ ′m)|S = sm : Sel(E/K,E[pN ])−→⊕`|m

H1`, f (K) . (3)

Since H1`, f (K) and H2

` (K) are Pontrjagin dual each other, we can take the dual basis t∗`,K of H1`, f (K) as

an RK/pN-module from the basis t`,K of H2` (K). Under the assumptions of Theorem 4.2.1, using the basis

t∗ì,K1≤i≤a of⊕a

i=1H1`, f (K), tì,K1≤i≤a of

⊕ai=1H

2ì(K) and the isomorphism Φ ′m, we have an exact

sequence⊕

`|mH1`, f (K)

f−→⊕a

i=1H2ì(K)−→ Sel(E/K,E[pN ])∨ −→ 0. Then the matrix corresponding to

f is an organizing matrix in the sense of Mazur and Rubin [15] (cf. [12] §9).

5 Modified Kolyvagin systems and numerical examples

5.1 Modified Kolyvagin systems of Gauss sum type

In §3.4 we constructed Kolyvagin systems κm,` for (m, `) such that m`∈N(N)1 (K[ε(m`)+1]). But the condition

` ∈ P(N)1 (K[ε(m`)+1]) is too strict, and it is not suitable for numerical computation. In this subsection, we

define a modified version of Kolyvagin systems of Gauss sum type for (m, `) such that m` ∈N(N)1 (K).

Suppose that K is in K(p). For each ` ∈ P(N)1 (K), we fix t` ∈ H0(F`,E[pN ]) of order pN , and consider

t`,K ∈H2` (K), whose `K-component is t`⊗ζ

⊗(−1)pN and other components are zero. Using t`,K , we regard ∂`

and φ` as homomorphisms ∂` : H1(K,E[pN ])−→ RK/pN and φ` : H1(K,E[pN ])−→ RK/pN .We will define an element κ

q,q′,zm,` in Sel(OK [1/m`],E[pN ]) for (m, `) such that m` ∈ N1(K) (and for

some primes q, q′ and some z in Sel(OK [1/qq′],E[pN ])). Consider (m, `) such that ` is a prime and m` ∈N1(K). We take n0 sufficiently large such that every prime of Kn0 dividing m` is inert in K∞/Kn0 . Then byProposition 3.3.2 (1), for any q ∈ (P′0)

(N)(K(m`)[1]Kn0+N), κm`,q ∈ Sel(OK [1/m`q],E[pN ]) satisfies

∂r(κm`,q) = φr(κ m`r ,q)

for all r dividing m`. By Lemma 3.4.1, we can take q, q′ ∈ (P′0)(N)(K(m`)[1]Kn0+N) satisfying

• wK(tq,K) = wK(tq′,K), and• there is z ∈ H1

f (OK [1/qq′],E[pN ]) such that ∂K(z) = tq,K − tq′,K , φ`(z) = t`,K and φr(z) = 0 for any rdividing m.

For any m ∈N1(K), let δm be the element defined in (3). We define

κq,q′,zm,` = κm`,q−κm`,q′ −δm`z . (1)


By Proposition 3.3.2 (2), we have κq,q′,zm,` ∈ Sel(OK [1/m`],E[pN ]).

Proposition 5.1.1 (0) κq,q′,zm,` is in Sel(OK [1/m`],E[pN ]).

(1) The element κq,q′,zm,` satisfies ∂r(κ

q,q′,zm,` ) = φr(κ

q,q′,zmr ,`

) for any prime divisor r of m.(2) We further assume that m` is admissible in the sense of the paragraph before Proposition 3.3.2. Thenwe have φr(κ

q,q′,zm,` ) = 0 for any prime divisor r of m.

(3) Under the same assumptions as (2), φ`(κq,q′,zm,` ) =−δm` holds.

Proof. (1) Using the definition of κq,q′,zm,` and Proposition 3.3.2 (1), we have ∂r(κ

q,q′,zm,` ) = ∂r(κm`,q−κm`,q′) =

φr(κ m`r ,q−κ m`

r ,q′). Next, we use the definition of κq,q′,zmr ,`

and φr(z) = 0 to get φr(κ m`r ,q−κ m`

r ,q′) = φr(κq,q′,zmr ,`

+

δ m`r

z) = φr(κq,q′,zmr ,`

). These computations imply (1).(2) We have φr(κm`,q) = φr(κm`,q′) = 0 by Proposition 3.3.2 (3). This together with φr(z) = 0 implies

φr(κq,q′,zm,` ) = φr(κm`,q−κm`,q′ −δm`z) = 0.

(3) We again use Proposition 3.3.2 (3) to get φ`(κm`,q) = φ`(κm`,q′) = 0. Since φ`(z) = 1, we have

φ`(κq,q′,zm,` ) = φ`(κm`,q−κm`,q′ −δm`z) =−δm`. This completes the proof of Proposition 5.1.1.

5.2 Proof of Theorem 1.2.5

In this subsection we take K = Q. For m ∈N(N) =N(N)(Q), we consider δm ∈Z/pN , which is defined fromϑQ(m) by (3). We define δm ∈ Z/pN by

θQ(m) ≡ δm

r

∏i=1

(τ`i −1) (mod pN ,(τ`1 −1)2, ...,(τ`r −1)2) (1)

where m = `1 · ... ·`r. By (4), θQ(m) = uϑQ(m) for some unit u∈ R×Q(m). This together with (3) and (1) implies

thatordp(δm) = ordp(δm). (2)

We take a generator η` ∈ (Z/`Z)× such that the image of ση`∈Gal(Q(µ`)/Q)' (Z/`)× in Gal(Q(`)/Q)'

(Z/`)×⊗Zp is τ` which is the generator we took. Then, using (1) and (1), we can easily check that theequation (2) in §1 holds.

In the rest of this subsection, we take N = 1. We simply write P1 for P(1)1 , so

P1 = ` ∈ Pgood | `≡ 1 (mod p) and E(F`)' Z/p.

The set of squarefree products of primes in P1 is denoted by N1.We first prove the following lemma which is related to the functional equation of an elliptic curve.

Lemma 5.2.1 Let ε be the root number of E. Suppose that m ∈ N1 is δ -minimal (for the definition ofδ -minimalness, see the paragraph before Conjecture 1.2.4). Then we have ε = (−1)ε(m).

Proof. By the functional equation (1.6.2) in Mazur and Tate [16] and the above definition of δm, we haveε(−1)ε(m)δm ≡ δm (mod p). Since δm 6≡ 0 (mod p) is equivalent to δm 6≡ 0 (mod p) by (2), we get theconclusion.

For each `∈P1, we fix a generator t` ∈H2` (Q) = H0(F`,E[p](−1))'Z/p = Fp, and regard φ` as a map

φ` : H1(Q,E[p])−→ Fp. Note that the restriction of φ` to Sel(E/Q,E[p]) is the zero map if and only if thenatural map s` : Sel(E/Q,E[p])−→ E(F`)⊗Z/p' Fp is the zero map.

I) Proof of Theorem 1.2.5 (1), (2).Suppose that ε(m)= 0, namely m= 1. Then δ1 = θQ mod p =L(E,1)/Ω

+E mod p. If δ1 6= 0, Sel(E/Q,E[p])=

0 and s1 is trivially bijective. Suppose next ε(m) = 1, so m = ` ∈ P1. It is sufficient to prove the next twopropositions.

246 Masato Kurihara

Proposition 5.2.2 Assume that ` ∈ P1 is δ -minimal. Then Sel(E/Q,E[p]) is 1-dimensional over Fp, ands` : Sel(E/Z,E[p]) −→ Fp is bijective. Moreover, the Selmer group Sel(E/Q,E[p∞])∨ with respect to thep-power torsion points E[p∞] is a free Zp-module of rank 1, namely Sel(E/Q,E[p∞])∨ ' Zp.

Proof. We first assume Sel(E/Q,E[p]) = 0 and will obtain the contradiction. We consider κq,q′,z1,` = κ`,q−

κ`,q′−δ`z, which was defined in (1). By Proposition 3.3.2 (1), we know ∂`(κq,q′,z1,` ) = φ`(gq−gq′). Consider

the exact sequence (see Lemma 3.2.1)

0−→ Sel(E/Q,E[p])−→ Sel(Z[1/r],E[p])−→H2r (Q)

for any r∈P1 where Sel(Z[1/r],E[p])−→H2r (Q)'Fp is nothing but ∂r. Since we assumed Sel(E/Q,E[p])=

0, Sel(Z[1/r],E[p]) −→H2r (Q) ' Fp is injective for any r ∈ P1. So ∂q(gq) = δ1 = 0 implies that gq = 0.

By the same method, we have gq′ = 0. Therefore, ∂`(κq,q′,z1,` ) = φ`(gq − gq′) = 0, which implies that

κq,q′,z1,` ∈ Sel(E/Q,E[p]).

But Proposition 5.1.1 (3) tells us that φ`(κq,q′,z1,` ) =−δ` 6= 0. Therefore, κ

q,q′,z1,` 6= 0, which contradicts our

assumption Sel(E/Q,E[p]) = 0. Thus we get Sel(E/Q,E[p]) 6= 0.On the other hand, by Corollary 4.1.3 we know that s` : Sel(E/Q,E[p]) −→ Fp is injective, therefore

bijective.By Lemma 5.2.1, the root number ε is −1. This shows that Sel(E/Q,E[p∞])∨ has positive Zp-rank by

the parity conjecture proved by Nekovár ([Nek06]). Therefore, we finally have Sel(E/Q,E[p∞])∨ ' Zp,which completes the proof of Proposition 5.2.2.

If we assume a slightly stronger condition on `, we also obtain the main conjecture. Let λ ′ = λ an be theanalytic λ -invariant of the p-adic L-function ϑQ∞

. We put nλ ′ = minn ∈ Z | pn−1≥ λ ′.

Proposition 5.2.3 Suppose that there is ` ∈ P1 such that

`≡ 1 (mod pnλ ′+2) and δ` 6= 0.

Then the main conjecture for (E,Q∞/Q) is true and Sel(E/Q∞,E[p∞])∨ is generated by one element as aΛQ∞

-module.

Proof. We use our Euler system g(K)` in §3.2 instead of κ

q,q′,z1,` which was used in the proof of Proposition

5.2.2. Let λ be the algebraic λ -invariant, namely the rank of Sel(E/Q∞,E[p∞])∨. Then λ ≤ λ ′ and ϑQ∞∈

char(Sel(OQ∞,E[p∞])∨) by Kato’s theorem.

Put K = Qnλ ′ and f = pn

λ ′ . Consider the group ring RK/p = Fp[Gal(K/Q)]. We identify a generatorγ of Gal(K/Q) with 1+ t, and identify RK/p with Fp[[t]]/(t

f ). The norm NGal(K/Q) = Σf−1

i=0 γ i is t f−1 bythis identification, so our assumption λ ′ ≤ f − 1 implies that the corestriction map Sel(E/K,E[p]) −→Sel(E/Q,E[p]) is the zero map because λ ≤ λ ′. Therefore, we have Q[1] ⊂ K. Since pn

λ ′+1− pnλ ′ > pn

λ ′ −1≥ λ ′ ≥ λ , the corestriction map Sel(E/Qn

λ ′+1,E[p])−→ Sel(E/Qnλ ′ ,E[p]) = Sel(E/K,E[p]) is also the

zero map. This shows that Q[2] ⊂Qnλ ′+1.

Our assumption `≡ 1 (mod pnλ ′+2) implies that ` splits completely in Qn

λ ′+1, so we have `∈P1(Q[2]) =P1(K[1]). Therefore, we can define

g(K)` ∈ Sel(OK [1/`],E[p])

in §3.2. Since ` ∈ P1(Q[2]), we also have

φ`(g(Q)` ) =−δ

(Q)` =−δ`

by Proposition 3.4.2 (4). It follows from our assumption δ` 6= 0 that g(Q)` 6= 0. Since CorK/Q(g

(K)` ) = g(Q)

`and the natural map i : Sel(Z[1/`],E[p])−→ Sel(OK [1/`],E[p]) is injective, we get

i(g(Q)` ) = NGal(K/Q)g

(K)` = t f−1g(K)

` 6= 0.

Consider ∂` : Sel(OK [1/`],E[p]) −→ RK/p. By definition, we have ∂`(g(K)` ) = utλ

′for some unit u

of RK/p. This shows that ∂`(tf−λ ′g(K)

` ) = 0, which implies that t f−λ ′g(K)` ∈ Sel(E/K,E[p]). The fact

t f−1g(K)` 6= 0 implies the submodule generated by t f−λ ′g(K)

` is isomorphic to RK/(p, tλ′) as an RK-module.


Namely, we haveSel(E/K,E[p])⊃ 〈t f−λ ′g(K)

` 〉 ' RK/(p, tλ′).

This implies that λ = λ ′, and Sel(E/K,E[p]) ' RK/(p, tλ ). Therefore, we have Sel(E/Q∞,E[p])∨ 'ΛQ∞

/(p,ϑQ∞). This together with Kato’s theorem we mentioned implies that Sel(E/Q∞,E[p∞])∨'ΛQ∞

/(ϑQ∞).

II) Proof of Theorem 1.2.5 (3).Suppose that m = `1`2 ∈ N1 and m is δ -minimal. As in the proof of Proposition 5.2.2, we assumeSel(E/Q,E[p]) = 0 and will get the contradiction. We consider κ

q,q′,z`1,`2

defined in (1). Consider the exactsequence (see Lemma 3.2.1)

0−→ Sel(E/Q,E[p])−→ Sel(Z[1/`1`2qq′],E[p]) ∂−→⊕

v∈`1,`2,q,q′H2

v(Q).

By the same method as the proof of Proposition 5.2.2, gq = gq′ = 0. Therefore, ∂`1(κ`1,q−κ`1,q′) = φ`1(gq−gq′) = 0 by Proposition 3.3.2 (1). We have ∂q(κ`1,q) = δ`1 = 0, ∂q(κ`1,q′) = 0, ∂q′(κ`1,q) = 0, ∂q′(κ`1,q′) =δ`1 = 0. Therefore, ∂ (κ`1,q−κ`1,q′) = 0. This together with Sel(E/Q,E[p]) = 0 shows that κ`1,q−κ`1,q′ = 0.Therefore, using Proposition 3.3.2 (1), we have

∂`2(κq,q′,z`1,`2

) = ∂`2(κm,q−κm,q′) = φ`2(κ`1,q−κ`1,q′) = 0.

By the same method as the above proof of κ`1,q − κ`1,q′ = 0, we get κq,q′,z1,`2

= 0. This implies that

∂`1(κq,q′,z`1,`2

) = φ`1(κq,q′,z1,`2

) = 0 by Proposition 5.1.1 (1). It follows that ∂ (κq,q′,z`1,`2

) = 0, which implies κq,q′,z`1,`2

∈Sel(E/Q,E[p]). But this is a contradiction because we assumed Sel(E/Q,E[p]) = 0 and

φ`2(κq,q′,z`1,`2

) =−δm 6= 0

by Proposition 5.1.1 (3). Thus, we get Sel(E/Q,E[p]) 6= 0.Now the root number is 1 by Lemma 5.2.1, therefore, by the parity conjecture proved by Nekovár

([Nek06]), we obtain dimFp Sel(E/Q,E[p]) ≥ 2. On the other hand, by Corollary 4.1.3 we know thatsm : Sel(E/Q,E[p]) −→ (Fp)

⊕2 is injective. Therefore, the injectivity of sm implies the bijectivity of sm.This completes the proof of Theorem 1.2.5 (3).

We give a simple corollary.

Corollary 5.2.4 Suppose that there is m ∈N1 such that m is δ -minimal and ε(m) = 2. We further assumethat the analytic λ -invariant λ ′ is 2. Then the main conjecture for (E,Q∞/Q) holds.

Proof. Put t= γ−1 and identify ΛQ∞/p with Fp[[t]]. Let A be the relation matrix of S= Sel(E/Q∞,E[p∞])∨.

Since S/(p, t) = Sel(E/Q,E[p])∨ ' Fp⊕Fp, t2 divides detA mod p. Therefore, the algebraic λ -invariantis also 2. This implies the main conjecture because detA divides ϑQ∞

in ΛQ∞(Kato [7]).

III) Proof of Theorem 1.2.5 (4).

Lemma 5.2.5 Suppose that `, `1, `2 are distinct primes in P1 satisfying δ` = δ``1 = δ``2 = 0. Assume alsothat s` : Sel(E/Q,E[p]) −→ Fp is bijective, and that ``1, ``2 are both admissible. We take q, q′ such that

they satisfy the conditions when we defined κq,q′,z`1`2,`

. Then we have(1) Sel(E/Q,E[p]) = Sel(Z[1/`],E[p]),(2) κ

q,q′,z`1,`

= 0, κq,q′,z`2,`

= 0, and

(3) κq,q′,z`1`2,`

∈ Sel(E/Q,E[p]).

Proof. (1) Since s` is bijective, taking the dual, we get the bijectivity of H2` (Q) −→ Sel(E/Q,E[p])∨ =

Sel(Z,E[p])∨. By the exact sequence

0−→ Sel(Z,E[p])−→ Sel(Z[1/`],E[p]) ∂`−→H2` (Q)−→ Sel(Z,E[p])∨ −→ 0

in Lemma 3.2.1, we get Sel(E/Q,E[p]) = Sel(Z,E[p]) = Sel(Z[1/`],E[p]).

248 Masato Kurihara

(2) We first note that the bijectivity of s` : Sel(E/Q,E[p])−→Fp implies the bijectivity of φ` : Sel(E/Q,E[p])−→Fp. Since ∂q(κ`,q) = δ` = 0, κ`,q ∈ Sel(Z[1/`],E[p]) = Sel(E/Q,E[p]) where we used the property (1)which we have just proved. Proposition 3.3.2 (3) implies φ`(κ`,q) = 0, which implies κ`,q = 0 by the bijec-tivity of φ`. By the same method, we have κ`,q′ = 0. Therefore, we have

κq,q′,z1,` = κ`,q−κ`,q′ −δ`z = 0.

Therefore, Proposition 5.1.1 (1) implies ∂`1(κq,q′,z`1,`

)= φ`1(κq,q′,z1,` )= 0. This implies κ

q,q′,z`1,`∈Sel(Z[1/`],E[p])=

Sel(E/Q,E[p]). Using Proposition 5.1.1 (3), we have

φ`(κq,q′,z`1,`

) =−δ``1 = 0,

which implies κq,q′,z`1,`

= 0 by the bijectivity of φ`. The same proof works for κq,q′,z`2,`

.

(3) It follows from Proposition 5.1.1 (1) and Lemma 5.2.5 (2) that ∂ì(κq,q′,z`1`2,`

) = φì(κq,q′,z`1`2ì

,`) = 0 for each i =

1, 2. This implies κq,q′,z`1`2,`

∈ Sel(Z[1/`],E[p]). Using Sel(Z[1/`],E[p]) = Sel(E/Q,E[p]) which we provedin (1), we get the conclusion. This completes the proof of Lemma 5.2.5.

We next prove Theorem 1.2.5 (4). Assume that m = `1`2`3 ∈ N1, m is δ -minimal, m is admissible, andsì : Sel(E/Q,E[p])−→ Fp is surjective for each i = 1, 2.

We assume dimFp Sel(E/Q,E[p])= 1 and will get the contradiction. By this assumption, sì : Sel(E/Q,E[p])−→Fp for each i = 1, 2 is bijective. This implies that φì : Sel(E/Q,E[p])−→ Fp for each i = 1, 2 is also bijec-tive. By Lemma 5.2.5 (3) we get κ

q,q′,z`2`3,`1

∈ Sel(E/Q,E[p]), taking q, q′ satisfying the conditions when we

defined this element. By Proposition 5.1.1 (3), we have φ`1(κq,q′,z`2`3,`1

) =−δm 6= 0, which implies κq,q′,z`2`3,`1

6= 0.

But by Proposition 5.1.1 (2), we have φ`2(κq,q′,z`2`3,`1

) = 0. This contradicts the bijectivity of φ`2 . Therefore, weobtain dimFp Sel(E/Q,E[p])> 1.

By Lemma 5.2.1 and our assumption that m is δ -minimal, we know that the root number ε is −1. Thisshows that dimFp Sel(E/Q,E[p]) ≥ 3 by the parity conjecture proved by Nekovár ([Nek06]). On the otherhand, Corollary 4.1.3 implies that dimFp Sel(E/Q,E[p])≤ 3 and sm : Sel(E/Q,E[p])−→ F⊕3

p is injective.Therefore, the above map sm is bijective. This completes the proof of Theorem 1.2.5 (4).

5.3 Numerical examples

In this section, we give several numerical examples.Let E = X0(11)(d) be the quadratic twist of X0(11) by d, namely dy2 = x3−4x2−160x−1264. We take

p = 3. Then if d ≡ 1 (mod p), p is a good ordinary prime which is not anomalous (namely ap(= a3) forE satisfies ap 6≡ 1 (mod p)), and p = 3 does not divide Tam(E), and the Galois representation on T3(E) issurjective. In the following examples, we checked µ ′ = 0 where µ ′ is the analytic µ-invariant. Then thisimplies that the algebraic µ-invariant is also zero (Kato [7] Theorem 17.4 (3)) under our assumptions. Inthe computations of δm below, we have to fix a generator of Gal(Q(`)/Q) ' (Z/`Z)× for a prime `. Wealways take the least primitive root η` of (Z/`Z)×. We compute δm using the formula in (2).

(1) d = 13. We take N = 1. Since δ7 = 20 6≡ 0 (mod 3), we know that Fitt1,F3(Sel(E/Q,E[3])∨) = F3 byTheorem 2.4.1, so Sel(E/Q,E[3]) is generated by one element.

The root number is ε = ( 1311 ) =−1, so L(E,1) = 0. We compute P1 = 7,31,73, .... Therefore, δ7 6≡ 0

(mod 3) implies Sel(E/Q,E[3])' F3 and

Sel(E/Q,E[3∞])∨ ' Z3

by Proposition 5.2.2. Also, it is easily computed that λ ′= 1 in this case. This implies that Sel(E/Q∞,E[3∞])∨'Z3, so the main conjecture also holds.

We can find a point P = (7045/36,−574201/216) of infinite order on the minimal Weierstrass modely2 +y = x3−x2−1746x−50295 of E = X0(11)(13). Therefore, we know X(E/Q)[3∞] = 0. We can easilycheck that E(F7) is cyclic of order 6, and that the image of the point P in E(F7)/3E(F7) is non-zero. So


we also checked numerically that s7 : Sel(E/Q,E[3]) −→ E(F7)/3E(F7) is bijective as Proposition 5.2.2claims.

(2) d = 40. We know ε =( 4011 )=−1. We take N = 1. We can compute P1 = 7,67,73, ..., and δ7 =−40 6≡ 0

(mod 3). This implies that Sel(E/Q,E[3])' F3 and Sel(E/Q,E[3∞])∨ ' Z3 by Proposition 5.2.2.In this case, we know λ ′ = 7. Therefore, nλ ′ = 2. We can check 5347 ∈ P1 (where 5347 ≡ 1 (mod 35))

and δ5347 =−412820 6≡ 0 (mod 3). Therefore, the main conjecture holds by Proposition 5.2.3. In this case,we can check that the p-adic L-function ϑQ∞

is divisible by (1+ t)3−1, so we have

rankZ3 Sel(E/Q1,E[3∞])∨ = 3

where Q1 is the first layer of Q∞/Q.

In the following, for a prime `∈P, we take a generator τ` of Gal(Q(`)/Q)' (Z/`Z)× and put S = τ`−1.We write ϑQ(`) = Σa(`)i Si where a(`)i ∈ Zp. Note that δ` = a(`)1 .

(3) d = 157. We know ε = ( 15711 ) = 1 and L(E,1)/Ω

+E = 45. We take N = 1. We compute a(37)

2 =

−14065/2 6≡ 0 (mod 3). Since 37≡ 1 (mod 32), c2 = 2−1 = 1 and a(37)2 is in Fitt2,F3(Sel(E/Q,E[3])∨) by

Theorem 2.4.1, which implies that Fitt2,F3(Sel(E/Q,E[3])) = F3. Therefore, Sel(E/Q,E[3]) is generatedby at most two elements.

We compute P1 = 7,67,73,127, .... Since 127 ≡ 1 (mod 7), 7× 127 is admissible. We computeδ7×127 = 83165 6≡ 0 (mod 3). Therefore, 7× 127 is δ -minimal. It follows from Theorem 1.2.5 (3) thatSel(E/Q,E[3])' F3⊕F3. In this example, we can check λ ′ = 2, so Corollary 5.2.4 together with the abovecomputation implies the main conjecture. Since L(E,1)/Ω

+E = 45 6= 0, rankE(Q) = 0 by Kato, which im-

plies Sel(E/Q,E[3∞]) =X(E/Q)[3∞]. Since 45∈ Fitt0,Z3(Sel(E/Q,E[3∞])∨), we have #X(E/Q)[3∞]≤9, and

X(E/Q)[3∞]' Z/3Z⊕Z/3Z.

(4) d = 265. In this case, ε = ( 26511 ) = 1 and L(E,1) = 0. We take N = 1. As in Example (3), we com-

pute a(37)2 = 16985 6≡ 0 (mod 3), which implies that Sel(E/Q,E[3]) is generated by at most two ele-

ments as above. We compute P1 = 7,13,31,67,103,109,127, .... For an admissible pair 7,127, wehave δ7×127 = −138880 6≡ 0 (mod 3). Therefore, 7× 127 is δ -minimal and Sel(E/Q,E[3]) ' F3⊕F3 byTheorem 1.2.5 (3). Since λ ′ = 2 in this case, by Corollary 5.2.4 we know that the main conjecture holds.

Since L(E,1) = 0, we know rankSel(E/Q,E[3∞])∨ > 0 by the main conjecture. This implies that

Sel(E/Q,E[3∞])∨ ' Z3⊕Z3.

Now E has a minimal Weierstrass model y2 + y = x3− x2−725658x−430708782. We can find rationalpoints P = (2403,108146) and Q = (5901,−448036) on this curve. We can also easily check that E(F7)is cyclic group of order 6 and E(F31) is cyclic of order 39. The image of P in E(F7)/3E(F7) ' Z/3Z is 0(the identity element), and the image of Q in E(F7)/3E(F7) ' Z/3Z is of order 3. On the other hand, theimages of P and Q in E(F31)/3E(F31) ' Z/3Z do not vanish and coincide. This shows that P and Q arelinearly independent over Z3. Therefore,

rankE(Q) = 2 and X(E/Q)[3∞] = 0.

In the above argument we considered the images of E(Q) in E(F7)/3E(F7) and E(F31)/3E(F31). Whatwe explained above implies that the natural map s7×31 : E(Q)/3E(Q)−→E(F7)/3E(F7)⊕E(F31)/3E(F31)is bijective. In this example, δ7×31 =−15290 6≡ 0 (mod 3), so Conjecture 1.2.4 holds for m = 7×31.

(5) d = 853. We know ε = ( 85311 ) =−1. Take N = 1 at first. For `= 271, we have a(271)

3 = 900852395/2 6≡ 0(mod 3), which implies that dimF3 Sel(E/Q,E[3]) ≤ 3. We compute P1 = 7,13,67,103,109, ...,463, ....We can find a rational point P = (1194979057/51984,40988136480065/11852352) on the minimal Weier-strass equation y2 + y = x3 − x2 − 7518626x− 14370149745 of E = X0(11)(853). We know that E(F7)is cyclic of order 6, and E(F13) is cyclic of order 18. Both of the images of P in E(F7)/3E(F7) andE(F13)/3E(F13) are of order 3. Therefore, s` : Sel(E/Q,E[3]) −→ E(F`)/3E(F`) is surjective for each` = 7, 13. Since 13 = −1 ∈ (F×7 )

3, 463 = 1 ∈ (F×7 )3 and 463 = 8 ∈ (F×13)

3, 7,13,463 is admissible. We

250 Masato Kurihara

can compute δ7×13×463 = −8676400 6≡ 0 (mod 3), and can check that m = 7×13×463 is δ -minimal. ByTheorem 1.2.5 (4), we have

Sel(E/Q,E[3])' F3⊕F3⊕F3. (1)

We have a rational point P of infinite order, so the rank of E(Q) is≥ 1. Take N = 3 and consider `= 271.Since δ271 = a(271)

1 = 35325 ≡ 9 (mod 27), 9 is in Fitt1,Z/p3Z(Sel(E/Q,E[33])∨) by Corollary 2.4.2. Thisimplies that rankE(Q) = 1 and #X(E/Q)[3∞]≤ 9. This together with (1) implies that

X(E/Q)[3∞]' Z/3Z⊕Z/3Z. (2)

Note that if we used only Theorem 1.1.1 and these computations, we could not get (1) nor (2) becausewe could not determine Θ1(Q)(δ ) by finite numbers of computations. We need Theorem 1.2.5 to obtain (1)and (2).

(6) For positive integers d which are conductors of even Dirichlet characters (so d = 4m or d = 4m+1 forsome m) satisfying 1≤ d ≤ 1000, d ≡ 1 (mod 3), and d 6≡ 0 (mod 11), we computed Sel(E/Q,E[3]). ThendimSel(E/Q,E[3]) = 0,1,2,3, and the case of dimension = 3 occurs only for d = 853 in Example (5).

(7) We also considered negative twists. Take d =−2963. In this case, we know L(E,1) 6= 0 and L(E,1)/Ω+E =

81. We know from the main conjecture that the order of the 3-component of X(E/Q) is 81, but the mainconjecture does not tell the structure of this group. Take N = 1 and ` = 19. Then we compute a(19)

2 =2753/2 6≡ 0 (mod 3) (we have ϑQ(19) ≡ −432S+(2753/2)S2 mod (9,S3)). Since c2 = 1, this shows that

a(19)2 is in Fitt2,F3(Sel(E/Q,E[3])∨) by Theorem 2.4.1. Therefore, we have Fitt2,F3(Sel(E/Q,E[3])) = F3,

which implies that Sel(E/Q,E[3])' (F3)⊕2. This denies the possibility of X(E/Q)[3∞]' (Z/3Z)⊕4, and

we haveX(E/Q)[3∞]' Z/9Z⊕Z/9Z.

(8) Let E be the curve y2 + xy+ y = x3 + x2− 15x+ 16 which is 563A1 in Cremona’s book [1]. We takep = 3. Since a3 = −1, Tam(E) = 1, µ = 0 and the Galois representation on T3(E) is surjective, all theconditions we assumed are satisfied. We know ε = 1 and L(E,1) = 0. Take N = 1. We compute P1 =13,61,103,109,127,139, .... For admissible pairs 13,103, 13,109, we compute δ13×103 =−6819≡0 (mod 3) and δ13×109 =−242 6≡ 0 (mod 3). From the latter, we know that

s13×109 : Sel(E/Q,E[3]) '−→ (F3)⊕2

is bijective by Theorem 1.2.5 (3). Since λ ′ = 2, the main conjecture also holds by Corollary 5.2.4. We knowL(E,1) = 0, so Sel(E/Q,E[3∞])' (Z3)

⊕2.Numerically, we can find rational points P = (2,−2) and Q = (−4,7) on this elliptic curve. We can

check that E(F13) is cyclic of order 12, E(F103) is cyclic of order 84, and E(F109) is cyclic of order 102.The points P and Q have the same image and do not vanish in E(F13)/3E(F13), but the image of P inE(F109)/3E(F109) is zero, and the image of Q in E(F109)/3E(F109) is non-zero. This shows that P and Qare linearly independent over Z3, and s13×109 is certainly bijective. Since all the elements in Sel(E/Q,E[3∞])come from the points, we have X(E/Q)[3∞] = 0. On the other hand, the image of P in E(F103)/3E(F103)coincides with the image of Q, so s13×103 is not bijective. This is an example for which δ13×103 ≡ 0 (mod3) and s13×103 is not bijective.

(9) Let E be the elliptic curve y2 + xy+ y = x3 + x2−10x+6 which has conductor 18097. We take p = 3.We know a3 = −1, Tam(E) = 1, µ = 0 and the Galois representation on T3(E) is surjective, so all theconditions we assumed are satisfied. In this case, ε = −1 and L(E,1) = 0. Take N = 1. We compute P1 =7,19,31,43,79, ...,601, .... We know 7,43,601 is admissible. We have δ7×43×601 = −2424748 6≡ 0(mod 3), and 7×43×601 is δ -minimal. We thank K. Matsuno heartily for his computing this value for us.The group E(F7) is cyclic of order 9 and E(F43) is cyclic of order 42. The point (0,2) is on this elliptic curve,and has non-zero image both in E(F7)/3E(F7) and E(F43)/3E(F43). So both s7 and s43 are surjective, andwe can apply Theorem 1.2.5 (4) to get

s7×43×601 : Sel(E/Q,E[3]) '−→ (F3)⊕3

is bijective.


Numerically, we can find 3 rational points P = (0,2), Q = (2,−1), R = (3,2) on this elliptic curve,and easily check that the restriction of s7×43×601 to the subgroup generated by P, Q, R in Sel(E/Q,E[3])is surjective. Therefore, we have checked numerically that s7×43×601 is bijective. This also implies thatrankE(Q) = 3 since E(Q)tors = 0. Therefore, all the elements of Sel(E/Q,E[3∞]) come from the rationalpoints, and we have X(E/Q)[3∞] = 0.

5.4 A Remark on ideal class groups

We consider the classical Stickelberger element

θStQ(µm)

=m

∑a=1

(a,m)=1

(12− a

m)σ−1

a ∈Q[Gal(Q(µm)/Q)]

(cf. (1)). Let K = Q(√−d) be an imaginary quadratic field with conductor d, and χ be the corresponding

quadratic character. Let m be a squarefree product whose prime divisors ` split in K and satisfy `≡ 1 (modp). Using the above classical Stickelberger element, we define δ St

m,K by

δStm,K =−

md

∑a=1

(a,md)=1

amd

χ(a)(∏`|m

logF`(a))

(cf. (2)). We denote by ClK the class group of K, and define the notion “δ StK -minimalness" analogously.

We consider the analogue of Conjecture 1.2.4 for δ Stm,K and dimFp(ClK/p). Namely, we ask whether

dimFp(ClK/p) = ε(m) for a δ StK -minimal m. Then the analogue does not hold. For example, take K =

Q(√−23) and p = 3. We know ClK ' Z/3Z. Put `1 = 151 and `2 = 211. We compute δ St

`1,K= −270 ≡ 0

(mod 3), δ St`2,K

= −1272 ≡ 0 (mod 3), and δ St`1·`2,K

= −415012 ≡ 2 (mod 3). This means that `1 · `2 is δ StK -

minimal. But, of course, we know dimFp(ClK/p) = 1 < 2 = ε(`1 · `2).

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Masato KURIHARADepartment of Mathematics,Keio University,3-14-1 Hiyoshi, Kohoku-ku,Yokohama, 223-8522, [email protected]

p-adic integration on ray class groups and non-ordinaryp-adic L-functions

David Loeffler

Abstract We study the theory of p-adic finite-order functions and distributions on ray class groups of num-ber fields, and apply this to the construction of (possibly unbounded) p-adic L-functions for automorphicforms on Gl2 which may be non-ordinary at the primes above p. As a consequence, we obtain a “plus-minus” decomposition of the p-adic L-functions of automorphic forms for Gl2 over an imaginary quadraticfield with p split and Hecke eigenvalues 0 at the primes above p, confirming a conjecture of B.D. Kim.

1 Introduction

P-adic L-functions attached to various classes of automorphic forms over number fields are an importantobject of study in Iwasawa theory. In most cases, these p-adic L-functions are constructed by interpolatingthe algebraic parts of critical values of classical (complex-analytic) L-functions of twists of the automorphicform by finite-order Hecke characters of the number field.

When the underlying automorphic form is ordinary at p, or when the number field is Q, this p-adic inter-polation is very well-understood. However, the case of non-ordinary automorphic forms for number fieldsK 6= Q has received comparatively little study. This paper grew out of the author’s attempts to understandthe work of Kim [9], who considered the L-functions of weight 2 elliptic modular forms over imaginaryquadratic fields.

In this paper, we develop a systematic theory of finite-order distributions and p-adic interpolation on rayclass groups of number fields. Developing such a theory is a somewhat delicate issue, and appears to havebeen the subject of several mistakes in the literature to date; we hope this paper will go some way towardsresolving the confusion. We then apply this theory to the study of p-adic L-functions for automorphicrepresentations of Gl2 over a general number field, extending the work of Shai Haran [5] in the ordinarycase. Largely for simplicity we assume the automorphic representations have trivial central character andlowest cohomological infinity-type (as in the case of representations attached to modular elliptic curves overK).

Applying our theory to the case of K imaginary quadratic with p split in K, we obtain proofs of twoconjectures. Firstly, we prove a special case of a conjecture advanced by the author and Zerbes in [10],confirming the existence of two “extra” p-adic L-functions whose existence was predicted on the basis ofgeneral conjectures on Euler systems. Secondly, we consider the case when the Hecke eigenvalues at bothprimes above p are 0 (the “most supersingular” case); in this case Kim has predicted a decomposition of theL-functions analogous to the “signed L-functions” of Pollack [13] for modular forms. Using the additionalinformation arising from our two extra L-functions, we prove Kim’s conjecture, constructing four boundedL-functions depending on a choice of sign at each of the two primes above p.

David LoefflerMathematics Institute, University of Warwick, Coventry CV4 7AL, UKe-mail: [email protected]

253

[email protected]

254 David Loeffler

Acknlowedgements

I would like to thank B.D. Kim for sending me an early draft of his paper [9], together with many informativeremarks on its contents. I am also grateful to Matthew Emerton, Günter Harder, Mladen Dimitrov and SarahZerbes for interesting and helpful conversations on this topic; and to the anonymous referee for severalvaluable comments and corrections.

2 Integration on p-adic groups

In this section, we shall recall and slightly generalize some results concerning the space of locally analyticdistributions on an abelian p-adic analytic group (the dual space of locally analytic functions); in particu-lar, we are interested in when it is possible to uniquely interpolate a linear functional on locally constantfunctions by a locally analytic distribution satisfying some growth property.

In this section we fix a prime p and a coefficient field E, which will be a complete discretely-valuedsubfield of Cp, endowed with the valuation vp such that vp(p) = 1.

2.1 One-variable theory

We first recall the well-known theory of finite-order functions and distributions on the group Zp.For r ∈ R≥0, we define the space of order r functions Cr(Zp,E) as the space of functions Zp → E

admitting a local Taylor expansion of degree brc at every point with error term o(zr), cf. [3, §I.5]. Wewrite Dr(Zp,E), the space of order r distributions, for the dual of Cr(Zp,E) (the space of linear functionalsCr(Zp,E)→ E which are continuous with respect to the Banach space topology of Cr(Zp,E); see op.cit. forthe definition of this topology).

We may regard the space Cla(Zp,E) of locally analytic E-valued functions on Zp as a dense subspace ofCr(Zp,E) for any r, so dually all of the spaces Dr(Zp,E) may be regarded as subspaces of the E-algebraDla(Zp,E) of locally analytic distributions, the dual of Cla(Zp,E). It is well known that Dla(Zp,E) can beinterpreted as the algebra of functions on a rigid-analytic space over E (isomorphic to the open unit disc),whose points parametrize continuous characters of Zp.

It is clear that the above definitions extend naturally if Zp is replaced by any abelian p-adic analyticgroup of dimension 1, since any such group has a finite-index open subgroup H isomorphic to Zp; we say afunction, distribution, etc has a given property if its pullback under the map Zp ∼= a ·H ⊆ G has it for everya ∈ G/H.

We define a locally constant distribution to be a linear functional on the space LC(G,E) of locallyconstant functions on Zp. This is clearly equivalent to the data of a finitely-additive E-valued function onopen subsets of G, i.e. a “distribution” in the sense1 of some textbooks such as [14].

Definition 1. Let G be an abelian p-adic analytic group of dimension 1, and let µ be an E-valued locallyconstant distribution on G. We say µ has growth bounded by r if there exists C ∈ R such that

infa∈Zp

vp µ (1a+pmH)≥C− rm

for all m≥ 0, where H is some choice of subgroup of G isomorphic to Zp.

Remark 1. The definition is independent of the choice of H, although the constant C may not be so.

Theorem 1 (Amice–Vélu, Vishik). Let G be an abelian p-adic Lie group of dimension 1.

(i) If µ ∈ Dr(G,E), then the restriction of µ to LC(G,E) has growth bounded by r.

1 To avoid confusion we shall not use the term “distribution” alone in this paper, but rather in company with an adjectivalphrase, such as “locally analytic”, and generally an “X distribution” shall mean “a continuous linear functional on the space ofX functions” (with respect to some topology to be understood from the context).

Integration on ray class groups 255

(ii) Conversely, if r < 1 and µ is a locally constant distribution on G with growth bounded by r, there existsa unique element µ ∈ Dr(G,E) whose restriction to LC(G,E) is µ .

Proof. This is a special case of [3, Theorème II.3.2].

Remark 2. In the case r ≥ 1, one can check that any locally constant distribution µ with growth boundedby r can be extended to an element µ ∈ Dr(G,E), but this is only uniquely defined modulo `0Dr−1(G,E),where `0 is the order 1 distribution whose action on C1 functions is f 7→ f ′(0).

2.2 The case of several variables

We now consider the case of functions of several variables. First we consider the group G = Zdp, for some

integer d ≥ 1. Let r be a fixed real number; for simplicity, we shall assume that r < 1.

Definition 2. Let d ≥ 1 and let f : G→ E be any continuous function. For m≥ 0 define the quantity εm( f )by

εm( f ) = infx∈G,y∈pmG

vp [ f (x+ y)− f (x)] .

We say f has order r if εm( f )− rm→ ∞ as m→ ∞.

The space Cr(G,E) of functions with this property is evidently an E-Banach space with the valuation

vCr( f ) = inf(

infx∈G

vp( f (x)), infm≥0

(εm( f )− rm)

).

Remark 3. For d = 1 this reduces to the definition of the valuation denoted by v′Cr in [3]; in op.cit. thenotation vCr is used for another slightly different valuation on Cr(Zp,E), which is not equal to v′Cr butinduces the same topology.

If we define, for any f ∈Cr(G,E), a sequence of functions ( fm)m≥0 by

fm =pm−1

∑j1=0· · ·

pm−1

∑jd=0

f ( j1, . . . , jd)1( j1,..., jd)+pmG,

then it is clear from the definition of the valuation vCr that we have fm→ f (x) in the topology of Cr(G,E),so in particular the space LC(G,E) of locally constant E-valued functions on G is dense in Cr(G,E).

We define a space Dr(G,E) as the dual of Cr(G,E), as before.As in the case d = 1 all of these constructions are clearly local on G and thus extend2 to abelian groups

having an open subgroup isomorphic to Zdp. Since any abelian p-adic analytic group has this property for

some d ≥ 0, we obtain well-defined spaces Cr(G,E) and dually Dr(G,E) for any such group G.

Remark 4. This includes the case where G is finite, so d = 0; in this case we clearly have have Cr(G,E) =Cla(G,E) = Maps(G,E) and Dr(G,E) = Dla(G,E) = E[G] for any r ≥ 0.

We now consider the problem of interpolating locally constant distributions by order r distributions.

Definition 3. Let µ be a locally constant distribution on G. We say that µ has growth bounded by r if thereis a constant C and an open subgroup H of G isomorphic to Zd

p such that

infa∈G

vpµ (1a+pmH)≥C− rm

for all m≥ 0.

2 A little care is required here since for k ≥ 1 pullback along the inclusion Zp×pk

−→ Zp is a continuous map Cr(Zp,E)→Cr(Zp,E) but not an isometry if r 6= 0. So for a general G the space Cr(G,E) has a canonical topology, but not a canonicalvaluation inducing this topology.

256 David Loeffler

The natural analogue of the theorem of Amice–Vélu–Vishik is the following.

Theorem 2. Let G be an abelian p-adic analytic group, r ∈ [0,1), and µ a locally constant distribution onG with growth bounded by r. Then there is a unique element µ ∈ Dr(G,E) whose restriction to LC(Zp,E)is µ .

Proof. The proof of this result in the general case is very similar to the case d = 1, so we shall give onlya brief sketch. Clearly one may assume G = H = Zd

p. One first shows that the space of locally constantfunctions on Zd

p has a “wavelet basis” consisting of the indicator functions of the sets

S(i1, . . . , id) = (i1 + p`(i1)Zp)×·· ·× (id + p`(id)Zp)

for i1, . . . , id ≥ 0, where `(n) is as defined in [3, §1.3]. One then checks that the functions

e(i1,...,id),r := psup jbr`(i j)c1S(i1,...,id)

form a Banach basis of Cr(Zdp,E), so a linear functional on locally constant functions taking bounded values

on the e(i1,...,id),r extends uniquely to the whole of Cr(Zdp,E).

Remark 5. The case when G = OL, for some finite extension L/Qp, has been studied independently byDe Ieso [7], who shows the following more general result: a locally polynomial distribution of degree kwhich has growth of order r in a sense generalizing Definition 3 admits a unique extension to an element ofDr(G,E) if (and only if) r < k+1. This reduces to the theorem above for k = 0.

2.3 Groups with a quasi-factorization

The constructions we have used for Cr and Dr seem to be essentially the only sensible approaches to definingsuch spaces which are functorial with respect to arbitrary automorphisms of G. However, if one assumes alittle more structure on the group G then there are other possibilities, which allow a greater range of locallyconstant distributions to be p-adically interpolated.

Definition 4. Let G be an abelian p-adic analytic group, and let G be the Lie algebra of G. A quasi-factorization of G is a decomposition G=

⊕Hi of G as a direct sum of subspaces.

If (Hi) is a quasi-factorization of G, we can clearly choose (non-uniquely) closed subgroups Hi ⊆G suchthat Hi is the Lie algebra of Hi and ∏Hi is an open subgroup of G. We shall say that (Hi) are subgroupscompatible with the quasi-factorization (Hi).

Definition 5. Let G be an abelian p-adic analytic group with a quasi-factorization H = (H1, . . . ,Hg), andlet (H1, . . . ,Hg) be subgroups compatible with H. Let f : G→ E be continuous. Define

εm1,...,mg( f ) = infx∈G

y∈pm1 H1×···×pmg Hg

vp [ f (x+ y)− f (x)] .

We say f has order (r1, . . . ,rg), where ri ∈ [0,1), if we have

εm1,...,mg( f )− (r1m1 + · · ·+ rgmg)→ ∞ (1)

as (m1, . . . ,mg)→∞ (with respect to the filter of cofinite subsets of Ng, i.e. for any N there are finitely manyg-tuples (m1, . . . ,mg) such that the above expression is ≤ N)

We define C(r1,...,rg)(G,E) to be the space of functions of order (r1, . . . ,rg), equipped with the valuationgiven by

v(r1,...,rg)( f ) = inf[

infx∈G

vp( f (x)), inf(m1,...,mn)

(εm1,...,mg( f )− (r1m1 + · · ·+ rgmg)

)],

which makes it into an E-Banach space. We write D(r1,...,rg)(G,E) for its dual. It is clear that as topologicalvector spaces these do not depend on the choice of the (Hi).


Remark 6. One can check that if G = H1× ·· ·×Hg, then C(r1,...,rg)(G,E) is isomorphic to the completedtensor product ⊗

1≤i≤g

Cri(Hi,E).

The spaces C(r1,...,rg)(G,E) are clearly invariant under automorphisms of G preserving the quasi-factori-zation H.

Definition 6. Let G be an abelian p-adic analytic group with a quasi-factorization H = (H1, . . . ,Hg), andlet (H1, . . . ,Hg) be subgroups compatible with H. Let µ be a locally constant distribution on G.

We say µ has growth bounded by (r1, . . . ,rd) if the following condition holds: there is a constant C suchthat for all m1, . . . ,mg ∈ Z≥0, we have

infa∈G

vpµ

(1a+(pm1 H1)×···×(pmg Hg)

)≥C− (r1m1 + · · ·+ rgmg).

It is clear that whether or not µ has growth bounded by (ri) does not depend on the choice of (Hi),although the constant C will so depend.

Theorem 3. Suppose µ is a locally constant distribution with growth bounded by (r1, . . . ,rg), where ri ∈[0,1). Then there is a unique extension of µ to an element

µ ∈ D(r1,...,rg)(G,E).

Proof. As usual it suffices to assume that G = H1×·· ·×Hg. Then the isomorphism

C(r1,...,rg)(G,E)∼=⊗

1≤i≤g

Cri(Hi,E)

gives an explicit Banach space basis of C(r1,...,rg)(G,E), and the result is now clear.

Remark 7. The special case of Theorem 3 where G = Z2p, with its natural quasi-factorization, is studied in

§7.1 of [9], which was the inspiration for many of the results of this paper.

3 Distributions on ray class groups

We now consider the application of the above machinery to global settings. We give a fairly general formu-lation, in the hope that this theory will be useful in contexts other than those we consider in the sectionsbelow; sadly, this comes at the cost of somewhat cumbersome notation.

Let K be a number field. As usual, we define a “modulus” of K to be a finite formal product ∏Ni=1 vni

iwhere each vi is either a finite prime of K or a real place of K, and ni ∈ Z≥0, with ni ≤ 1 if vi is infinite. Wedefine a “pseudo-modulus” to be a similar formal product but with some of the ni at finite places allowedto be ∞. If F is a pseudo-modulus, then the ray class group of K modulo F is defined, and we denote thisgroup by GF.

Let F be a pseudo-modulus of K, and let Σ be the finite set of primes p of K such that p∞ | F.Let F be a finite extension of either Q or of Qp for some prime p, and V a finite-dimensional F-vector

space.

Definition 7. A growth parameter is a family δ = (δp)p∈Σ of non-zero elements of OF indexed by theprimes in Σ .

Definition 8. A ray class distribution modulo F with growth parameter δ and values in V to be the data of,for each modulus f dividing F, an element

Φf ∈ F [Gf]⊗F V,

such that:

258 David Loeffler

1. We have Normf′

f Φf′ = Φf for all pairs of moduli (f, f′) with f | f′ and f′ | F.2. There exists an OF -lattice Λ ⊆ V such that for every f | F, we have Φf ∈ OF [Gf]⊗OF δ

−1f Λ , where

δf = ∏p∈Σ δ−vp(f)p .

We shall see in subsequent sections that systems of elements of this kind appear in several contexts assteps in the construction of p-adic L-functions attached to automorphic forms.

Since the natural maps GF→ Gf for f | F are surjective, and GF is equal to the inverse limit of the finitegroups Gf over moduli f | F, condition (1) implies that we may interpret a ray class distribution as a locallyconstant distribution on GF with values in V . Condition (2) can then be interpreted as a growth conditionon this locally constant distribution. Our goal is to investigate how this interacts with the growth conditionsstudied in §2 above.

If the coefficient field F is a number field, then choosing a prime q of F allows us to interpret V -valued rayclass distributions as Vq-valued ray class distributions, where Vq = Fq⊗F V . So we shall assume henceforththat F = E is a finite extension of Qp for some rational prime p, as in §2.

Let us write F= g ·∏p∈Σ p∞, for some modulus g coprime to Σ . Then we have an exact sequence

0→ Eg→ ∏p∈Σ

O×K,p→ GF→ Gg→ 0

where Eg is the closure of the image in ∏p∈Σ O×K,p of the group of units of OK congruent3 to 1 modulo g.Since Gg is finite, we see that if Σ is a subset of the primes above p, then Gp∞g p-adic analytic group; andif

f= ∏p∈Σ

pnp

for some integers np, the kernel of the natural surjection GF→ Gfg is the image Hf of the subgroup

Uf := ∏p∈Σ

(1+pnpOK,p)× ⊆ ∏

p∈Σ

O×K,p.

The subgroups Hf for f | p∞ form a basis of neighbourhoods of 1 in Ggp∞ , so we may regard a ray classdistribution as a locally constant distribution on Ggp∞ satisfying a boundedness condition relative to thesubgroups Hf. Our task is to investigate how this interacts with the boundedness conditions considered in§2 above.

3.1 Criteria for distributions of order r

The following theorem gives a sufficient condition for a ray class distribution to define a finite-order distri-bution on the group GF in the sense of §2.2, when Σ is a subset of the of primes above p (so that GF is ap-adic analytic group). We continue to assume that the coefficient field E is a finite extension of Qp, and(as before) we write vp for the valuation on E such that vp(p) = 1. We shall assume for simplicity that thecoefficient space V is simply E; the general case follows immediately from this.

Theorem 4. Let Θ be a ray class distribution modulo F. Suppose Θ has growth parameter (αp)p∈Σ , anddefine

r = ∑p∈Σ

epvp(αp),

where ep is the absolute ramification index of the prime p. Then ΘF, viewed as a locally constant distributionon GF, has growth bounded by r in the sense of Definition 3.

In particular, if r < 1, there is a unique element Θ ∈ Dr(Gp∞ ,E) extending Θ .

Proof. Let p ∈ Σ . We can choose some integer c ≥ 0 such that the p-adic logarithm converges on Up,k =(1+ pmOp)

× ⊆ O×p for all k ≥ c, and identifies Up,k with the additive group pkOp. In particular, Up,c is

3 We adopt the usual convention in class field theory that if g is divisible by a real infinite place v, “congruent to 1 modulo v”means that the image of the unit concerned under the corresponding real embedding of K should be positive.


isomorphic to Z[Kp:Qp]p , and for m≥ 0 we have

U pm

p,c∼= pmpc

i Op = pc+epmOp∼=Up,c+epm.

Now let p1, . . . ,pg be the primes in Σ and choose a constant ci for each. Then the subgroup U =Up1,c1×·· ·×Upg,cg is an open subgroup of (OK⊗Zp)

×=∏gi=1O

×pi

, and U ∼=Z[K:Q]p . By increasing the ci if necessary,

we may assume that the image of U in GF is torsion-free and hence isomorphic to Zhp for some h≤ [K : Q].

Let H be the image of U . Then the image of U pmis H pm

, clearly; so a locally constant distribution onGF has order r if and only if there is C such that vpµ(1x·H pm ) ≥C− rm (this is just Definition 3, but withthe group law on GF written multiplicatively).

However, we have

U pm=

g

∏i=1

Upi,ci+epi m=

g

∏i=1

(1+pci+epi mi Opi)

×.

Thus GF/H pmis the ray class group of modulus

fm := g ·g

∏i=1

pci+epi mi ,

where g is (as above) the modulus such that F= g ·∏gi=1 p

∞i . So, if Θ is a ray class distribution with growth

parameter (αp)p∈Σ , then we know that the valuation of Θ(X) where X is any coset of H pmis bounded below

by

C−g

∑i=1

vp(αpi) ·ordpi(fm) =C′−mg

∑i=1

epivp(αpi) =C′− rm

for some constants C,C′, where r is as defined in the statement of the theorem. So a ray class distributionwith growth parameter (αp)p∈Σ defines a locally constant distribution on the p-adic analytic group G whosegrowth is bounded by r, as required.

Note that this argument does not depend on the dimension of GF (which is useful, since this dimensiondepends on whether Leopoldt’s conjecture holds for K). This result is essentially the best possible (at leastusing the present methods) when there is a unique prime of K above p, or when K is totally real andLeopoldt’s conjecture holds (as we show in the next section). However, for other fields K finer statementsare possible using the theory of quasi-factorizations developed in §2.3, as we shall see below.

3.2 A converse result for totally real fields

Let us now suppose K is totally real. We also suppose that Leopoldt’s conjecture holds for K, so the imageof E in ∏p|pO

×p has rank n−1. We shall take F= p∞g for some modulus g coprime to p, so Σ = Σp is the

set of all primes dividing p. Leopoldt’s conjecture implies that Gp∞g is a p-adic analytic group of dimension1; thus, in particular, the notions of finite-order functions and distributions on Gp∞g are just the standardones.

We shall prove the following theorem:

Theorem 5. Let α = (αp)p∈Σp be elements of E, and let h ∈ R≥0. Then the implication

“Every ray class distribution modulo p∞g of growth parameter α is a locally constant distribution on Gp∞g with growthbounded by h”

is true if and only if the inequality∑

p∈Σp

epvp(αp)≤ h

holds.

We retain the notation of the proof of Theorem 4, so p1, . . . ,pg are the primes above p, and for each i, ciis a constant such that for all k≥ ci the logarithm map identifies Upi,k with the additive group (pi)

m, and theimage of U =Up1,c1 × . . .Upg,cg in G is torsion-free and hence isomorphic to Zp.

260 David Loeffler

Enlarging the ci further if necessary, we may assume that there is an integer w independent of i such thatfor each i the image of pci

i under the trace map TrKpi/Qp is pwZp. Let di = [Kpi : Qp].

Proposition 1. For each i there exists a basis bi1, . . . ,bidi of pcii as a Zp-module such that

TrKpi/Qp(bi j) = pw

for each j.

Proof. Elementary linear algebra.

Proposition 2. The isomorphismU ∼= Zn

p

(where n = [K : Q] = ∑gi=1 di) given by the bases bi1, . . . ,bidi for 1 ≤ i ≤ g identifies the closure of E∩U

with the submodule

∆ = (x1, . . . ,xd) ∈ Zdp :

d

∑j=1

x j = 0.

Proof. With respect to the basis b1, . . . ,bei fi of pcii , the trace map is given by (x1, . . . ,xdi) 7→ pw

∑ j x j, soour isomorphism

Ulog−→

g

∏i=1

pcii∼=−→ Zn

p

identifies ∆ with the elements of U whose norm down to Q is 1. However, since U is torsion-free, anyu ∈U ∩E satisfies NormK/Q(u) = 1, so ∆ contains the closure of E∩U , which we write as ∆ ′.

By Leopoldt’s conjecture (which we are assuming), ∆ ′ has Zp-rank (n−1), the same as ∆ ; so the quotient∆

∆ ′ is finite. However, we obviously haveU∆ ′∼= Zp⊕

∆

∆ ′;

and U∆ ′ =U(p∞,pc1

1 . . .pcgg ) is torsion-free, so we must have ∆ = ∆ ′.

Proposition 3. Let H be the image of U in Gp∞g, and let Hm = H pmas above. Suppose m ≥ 1. Then if

r1, . . . ,rg are integers such that the image in G of Up1,r1 × ·· · ×Upg,rg is contained in Hm, we must haveri ≥ ci + eim for all i.

Proof. If the image of Up1,r1 × ·· · ×Upg,rg is contained in Hm, then the same must also be true with(r1, . . . ,rg) replaced by (r′1, . . . ,r

′g) where r′i = sup(ri,ci). So we may assume without loss of generality

that ri ≥ ci for all i. Now the result is clear from the above description of the image of the global units.

This result clearly implies Theorem 5.

Remark 8. Curiously, the results above seem to conflict with some statements asserted without proof in [12](and quoted by some other subsequent works). The group studied in op.cit. corresponds in our notation toGgp∞ , where g is the product of the infinite places, and K is assumed totally real. (Thus Ggp∞ correspondsvia class field theory to the Galois group of the maximal abelian extension of K unramified outside p andthe infinite places; it is denoted by Galp in op.cit..)

In Definition 4.2 of op.cit., a locally constant distribution µ on Ggp∞ is defined to be 1-admissible if

supa

∣∣µ (1a+(m)

)∣∣= o(|m|−1

p).

Paragraph 4.3 of op.cit. then claims that a 1-admissible measure extends uniquely to an element ofDla(Ggp∞ ,E) which, regarded as a rigid-analytic function on the space Xp parametrizing characters of Ggp∞ ,has growth o(log(1+X)).

This is consistent with Theorem 4 and Theorem 5 above if there is only one prime above p. However,if there are multiple primes above p, it is not so clear how |m| is to be defined for general ideals m | p∞ ofK, and the uniqueness assertion of Conjecture 6.2 of op.cit. (which is asserted to be a consequence of thetheory of h-admissible measures) contradicts Theorem 5 of the present paper.


3.3 Imaginary quadratic fields

Let K be an imaginary quadratic field. If p is inert or ramified in K then there is only one prime p of Kabove p, and hence there is no canonical direct sum decomposition of LieGp∞ ∼= Kp. On the other hand, wecan find one when p splits:

Proposition 4. If K is imaginary quadratic and p = pp is split, then the images of LieO×K,p and LieO×K,p inLie(Gp∞) form a quasi-factorization.

Proposition 5. Let µ be a ray class distribution modulo p∞g with growth parameter (αp,αp). Then µ hasgrowth bounded by (vp(αp),vp(αp)) in the sense of Definition 6.

Proof. Clear by construction.

Combining the above with Theorem 3 we have the following:

Theorem 6. Let K be an imaginary quadratic field in which p = pp is split, and let Θ be a ray classdistribution modulo p∞g with values in V and growth parameter (αp,αp). Let r = vp(αp) and s = vp(αp).If r,s < 1, then there is a unique distribution

Θ ∈ D(r,s)(Gp∞ ,E)⊗E V

such that the restriction of Θ to LC(Gp∞ ,E)⊗E V is Θ . If we have r+s< 1, then Θ lies in Dr+s(Gp∞ ,E)⊗E Vand agrees with the element constructed in Theorem 4.

3.4 Fields of higher degree

When K is a non-totally-real extension of degree > 2, one can sometimes find interesting quasi-factorizationsof Gp∞ by considering subsets of the primes above p. If we identify Gp∞ with the Galois group of the max-imal abelian extension K(p∞)/K unramified outside p, we seek to write K(p∞) as a compositum of smallerextensions, each ramified at as few primes as possible. We give a few examples of the behaviour that occurswhen p is totally split.

Example 1 (Mixed signature cubic fields). Suppose K is a non-totally-real cubic field, and p is totally split inK, say p = p1p2p3. Then Gp∞ has dimension 2 (since Leopoldt’s conjecture is trivially true in this case), andthe images of the groups O×K,pi

in Gp∞ give three pairwise-disjoint one-dimensional subspaces of LieGp∞ ,so any two of these form a quasi-factorization. One therefore obtains three quasi-factorizations of Gp∞ .

One checks that with respect to the quasi-factorization given by p1 and p2, a ray class distribution ofparameter (αp1 ,αp2 ,αp3) has growth bounded by (r,s) if and only if vp(αp1)+ vp(αp3)≤ r and vp(αp2)+vp(αp3) ≤ s. For instance, if all αi are equal and their common valuation h is < 1

2 , then can construct alocally analytic distribution (indeed three of them), while Theorem 4 would only apply if h < 1

3 . For thecases 1

3 ≤ h < 12 , it is not immediately obvious whether the interpolations provided by the three quasi-

factorizations are equal or not, but we shall see in the next section that this is indeed the case.

Example 2 (Quartic fields). Suppose K is a totally imaginary quartic field, and p is totally split in K. As inthe previous example, O×K has rank 1 and thus Leopoldt’s conjecture is automatic, so Gp∞ has rank 3. Wecan obtain a quasi-factorization of Gp∞ by considering the images of the Lie algebras of O×K,pi

for 1≤ i≤ 3;then (much as in the previous case) we find that the condition for a ray class distribution to have growthbounded by (r,s, t) is that

vp(α1)+ vp(α4)≤ r,

vp(α2)+ vp(α4)≤ s,

vp(α3)+ vp(α4)≤ t.

262 David Loeffler

A slightly more intricate class of quasi-factorizations can be obtained as follows. We can choose a basisfor LieO×K,pi

for each i such that the Lie algebra of the subgroup EK corresponds to the subspace ∆ of Q4p

spanned by (1,1,1,1). Then choosing a quasi-factorization amounts to choosing a basis of the 3-dimensionalspace of linear functionals on Q4

p which vanish on ∆ . One such basis is given by the functionals mapping(x1,x2,x3,x4) to x1 − x4,x2 − x4,x3 − x4, and this gives the quasi-factorization we have already seen.However, we can also consider the functionals x1−x2,x2−x3,x3−x4. One checks then that the conditionfor a ray class distribution to have growth bounded by (r,s, t) is

vp(α1)+ vp(α2)≤ r,

vp(α2)+ vp(α3)≤ s,

vp(α3)+ vp(α4)≤ t.

One can generalize the last example to a much wider range of number fields under suitable assumptionson the position of LieEK inside Lie(Zp⊗OK)

× (which amounts to assuming a strong form of Leopoldt’sconjecture, such as that formulated in [2]).

3.5 A representation-theoretic perspective

For simplicity let us suppose that K has class number 1 and g = (1), so G := Gp∞ = U/∆ where U =

∏gi=1O

×pi

and ∆ = O×K . Pick real numbers r1, . . . ,rg with ri ∈ [0,1)∩ vp(E×).We consider the inclusion

LC(U,E)∆ →C(r1,...,rg)(U,E)∆ ,

where we give U the obvious quasi-factorization. We have, of course, LC(U,E)∆ = LC(U/∆ ,E) =LC(G,E). The following proposition is immediate from the definitions:

Proposition 6. Let µ be an element of D(r1,...,rg)(U,E). Then the restriction of µ to LC(U,E)∆ is a rayclass distribution of parameter (α1, . . . ,αg), where the αi are any elements with vp(αi) = ri.

Conversely, any ray class distribution of parameter (α1, . . . ,αg) defines a linear functional on LC(U,E)∆

which is continuous with respect to the subspace topology given by the inclusion into C(r1,...,rg)(U,E).

However, even though LC(U,E) is dense in C(r1,...,rg)(U,E), it does not necessarily follow that LC(U,E)∆

is dense in C(r1,...,rg)(U,E)∆ ; the completion of the invariants may be smaller than the invariants of the com-pletion.

Example 3. Consider the case where U is Z2p with its natural quasi-factorization, and ∆ is the subgroup

(x,y) : x+ y = 0. Then a continuous function f : U → E is ∆ -invariant if and only if there is a functionh ∈ C0(Zp,E) such that f (x,y) = h(x+ y). So the coefficients in the Mahler–Amice expansion f (x,y) =∑m,n am,n

( xm

)(yn

)are given by am,n = bm+n, where h(x) = ∑n≥0 bn

(xn

). Since the functions

pbr1`(m)+r2`(n)c(

xm

)(yn

)are a Banach basis of C(r1,r2)(U,E), we see that f is in this space if and only if

limm,n→∞

vp(bm+n)− r1`(m)− r2`(n) = ∞.

The supremum of r1`(m)+ r2`(n) over pairs (m,n) with m+n = k is (r1 + r2)`(k)+O(1) as k→ ∞, sothis is equivalent to

limk→∞

vp(bk)− (r1 + r2)`(k) = ∞,

which is precisely the condition that h is Cr where r = r1 + r2. Moreover, this gives a Banach space isomor-phism between Cr(Zp,E) and C(r1,r2)(U,E)∆ preserving the subspaces of locally constant functions, so ifr1 + r2 ≥ 1 we see that LC(U,E)∆ is not dense in C(r1,r2)(U,E)∆ .


Remark 9. In §3.4, we gave criteria for locally constant functions on U/∆ to be dense in a Banach spacewhose topology is coarser than that of C(r1,...,rg)(U,E)∆ . This in particular implies that such functions aredense in C(r1,...,rg)(U,E)∆ in these cases, and hence that the multiple possible choices of auxilliary quasi-factorizations considered in the examples above do all give the same distribution on G when they apply.

4 Construction of p-adic L-functions

We now give the motivating example of a ray class distribution: the distribution constructed from the Mazur–Tate elements of an automorphic representation of Gl2 /K.

4.1 Mazur–Tate elements

Let K be a number field, and Π an automorphic representation of Gl2 /K. We make the following simplifyingassumptions:

1. Π is cohomological in trivial weight, i.e. the (g,K∞)-cohomology of Π does not vanish, where K∞ is amaximal compact connected subgroup of Gl2(K⊗R);

2. the central character of Π is trivial.

Conditions (1) and (2) are satisfied, for instance, if Π is the base-change to K of the automorphic rep-resentation of Gl2 /Q attached to a modular form of weight 2 and trivial nebentypus. It follows from (1)and (2) that there exists a finite extension F/Q inside C such that the Gl2(A∞

K)-representation Π ∞ can bedefined over F , and in particular the Hecke eigenvalues al(Π) for each prime l of K are in OF .

Theorem 7 (Haran). Under the above assumptions, there exists a finitely-generated OF -submodule ΛΠ ofC, and for each ideal f of K coprime to the conductor of Π an element

Θf(Π) ∈ Z[Gf]⊗Z ΛΠ ,

where Gf is the ray class group modulo f, such that the following relations hold:

(i) (Special values) If ω is a primitive ray class character of conductor f, then

ω(Θf(Π)

)=

Lf(Π ,ω,1)τ(ω) · |f|1/2 · (4π)[K:Q]

where Lf(Π ,ω,s) denotes the L-function of Π twisted by ω , without the Euler factors at ∞ or at primesdividing f, and τ(ω) is the Gauss sum (normalized so that |τ(ω)|= 1);

(ii) (Norm relation) For l a prime not dividing the conductor of Π , we have

Normflf Θfl(Π) =

(al(Π)−σl−σ

−1l

)Θf(Π) if l - f,

al(Π)Θf(Π)−Θf/l(Π) if l | f.

where σl denotes the class of l in Gf.

Proof. See [5]. (Note that in the main text of op.cit. it is assumed that K is totally imaginary, but thisassumption can easily be removed, as noted in section 7.)

Remark 10. In fact one can obtain essentially the same result under far weaker assumptions: it suffices tosuppose that Π is cuspidal, cohomological (of any weight), and critical. However, the necessary general-izations of Shai Haran’s arguments are time-consuming, so we shall not consider this more general settinghere. For the case when K is totally real, but Π has arbitrary critical weight and central character, see [4].

We refer to the elements Θf(Π)∈Z[Gf]⊗Z ΛΠ as Mazur–Tate elements, since they are closely analogousto the group ring elements considered in [11]. The coefficient module ΛΠ is in fact rather small, as thefollowing result (an analogue of the Manin–Drinfeld theorem) shows:

264 David Loeffler

Proposition 7. The OF -submodule ΛΠ ⊆ C may be taken to have rank 1.

Proof. We know that ΛΠ is finitely-generated, so it suffices to show that F⊗OF ΛΠ is 1-dimensional.To prove this, we must delve a little into the details of Shai Haran’s construction. The elements Θf(Π)

for varying Π are all obtained from a “universal Mazur–Tate element” in the module Z[Gf]⊗HBMr (Y,Z),

where Y is a locally symmetric space for Gl2(AK) and HBMr is Borel–Moore homology (the homological

analogue of compactly supported cohomology). Here r = r1 + r2, where r1 and r2 are the numbers of realand complex places of K. The element Θf(Π) is obtained by integrating the universal Mazur–Tate elementagainst a class in Hr

par(Y,C) arising from Π . However, it follows from the results of [6] that there is a uniqueHecke-invariant section of the projection map

Hrc (Y,C)→ Hr

par(Y,C),

and the Π -isotypical component of Hrc (Y,C) is 1-dimensional and descends to Hr

c (Y,F). Thus, after renor-malizing by a single (probably transcendental) complex constant depending on Π , the Mazur–Tate elementsfor Π can be obtained as values of the perfect pairing

HBMr (Y,F)×Hr

c (Y,F)→ F,

so we deduce that F⊗OF ΛΠ has dimension 1 as claimed.

4.2 Stabilization

We now introduce “Σ -stabilized” versions of the Θf(Π), again following [5] closely. Let F be a pseudo-modulus of K, and Σ the set of primes p such that p∞ | F, as before. We assume that none of the primesp ∈ Σ divide the conductor of Π .

Definition 9. An Σ -refinement of Π is the data of, for each p ∈ Σ , a root αp ∈ F of the polynomial

Pp(Π) = X2−ap(Π)X +NK/Q(p) ∈ F [X ]

(the local L-factor of Π at p).

We write βp for the complementary root to αp. By enlarging F if necessary, we may assume that the αp

and βp all lie in F .We introduce a formal operator Rp on the Θf(Π)’s by Rp ·Θf(Π) = Θf/p(Π) whenever p | f. Then it is

easy to see that the Rp for different p | f commute with each other, so we can define

ΘΣf (Π) = α

−1f

(∏p∈Σ

(1−βpRp)

)Θf(Π),

whenever f is divisible by all p ∈ Σ , where αf stands for the product

αf := ∏p∈Σ

α−vp(f)p .

Then one checks that we haveNormfp

f

(Θ

Σfp(Π)

)=Θ

Σf (Π)

for any p such that p | f and p ∈ Σ . We can therefore extend the definition of Θ Σf (Π) uniquely to all f | F in

such a way that this formula still holds.

Corollary 1. Let F be a modulus of K and write F= gF∞, where F= ∏p∈Σ p∞ and g is coprime to Σ . Thenthe elements

ΘΣfg(Π) : f | F∞

defined above form a ray class distribution modulo F, with values in F ⊗OF ΛΠ and growth parameter(αp)p∈Σ .


4.3 Consequences for p-adic L-functions

We now summarize the results on p-adic L-functions that can be obtained by applying our theory to the rayclass distributions constructed in §4.2. As before, let F be a number field, and E be the completion of F atsome choice of prime above p; and let vp denote the p-adic valuation on E normalized so that vp(p) = 1.

Theorem 8. Let K be an arbitrary number field and Σ a subset of the primes of K above p, and let F be thepseudo-modulus g ·∏p∈Σ p∞, where g is a modulus coprime to Σ . Let Π be an automorphic representationof GL2 /K with Hecke eigenvalues in F, satisfying the hypotheses (1) and (2) of §4.1, and unramified at theprimes in Σ . Let α = (αp)p∈Σ be a Σ -refinement of Π defined over F. If we have

h = ∑p∈Σ

epvp(αp)< 1,

then there is a unique distribution Lp,α(Π) ∈Dh(GF,E)⊗OF ΛΠ such that for all finite-order characters ω

of GF whose conductor is divisible by g, we have

Lp,α(Π)(ω) =

(∏p∈Σ

α−ordpfp

)·

(∏

p∈Σ ,p-f(1−αpω(p))(1−αpω(p)−1)

)

·Lf(Π ,ω,1)

τ(ω) · |f| · (4π)[K:Q], (1)

where f is the conductor of ω .

Proof. This is immediate from Theorem 7 and Corollary 1 except for the formula for Lp,α(Π)(ω) when fis not divisible by all primes in Σ ; the latter follows via a short explicit calculation from part (ii) of Theorem7.

If K is imaginary quadratic and p is split, we obtain a slightly finer statement using the quasi-factorizationabove:

Theorem 9. Let K be an imaginary quadratic field in which p = pp splits, and let Π be an automorphicrepresentation satisfying the hypotheses (1) and (2) of §4.1 with Hecke eigenvalues in F, and unramified atp and p. Let α = (αp,αp) be a p-refinement of Π defined over F.

If we haver = vp(αp)< 1 and s = vp(αp)< 1,

then there is a unique distribution Lp,α(Π) ∈ D(r,s)(Gp∞ ,E)⊗OF ΛΠ satisfying the interpolating property(1).

Note that every Π will admit at least one p-refinement satisfying the hypotheses of Theorem 9, andindeed if Π is non-ordinary at p and p then all four p-refinements do so; but at most two of the four p-refinements, and in many cases none at all, will satisfy the hypotheses of Theorem 8.

Remark 11. Applying Theorem 9 to the base-change of a classical modular form, we obtain Conjecture6.13 of our earlier paper [10] for all weight 2 modular forms whose central character is trivial.

Remark 12. Examples of ray class distributions appear to arise in more or less any context where p-adicinterpolation of L-values of non-ordinary automorphic representations over general number fields is con-sidered. For instance, the works of Ash–Ginzburg [1] on p-adic L-functions for Gl2n over a number field,and of Januszewski on Rankin–Selberg convolutions on Gln×Gln−1 over a number field [8], can both beviewed as giving rise to a ray class distribution (with some parameter depending on the Hecke eigenvaluesof the automorphic representations involved). Thus the theory developed in this paper allows these worksto be extended to non-ordinary automorphic representations of (sufficiently small) non-zero slope.

266 David Loeffler

5 The ap = 0 case

We now concentrate on a special case of Theorem 9, where K is imaginary quadratic with p split andap(Π) = ap(Π) = 0 (as in the case of the automorphic representation attached to a modular elliptic curvewith good supersingular reduction at the primes above p, assuming p≥ 5). Then the two Hecke polynomialsat p and p are the same, and we write α,β for their common set of roots, which are both square roots of−p and in particular have normalized p-adic valuation 1

2 . Thus Theorem 9 furnishes four p-adic L-functions.

Remark 13. None of these four L-functions can be obtained using Theorem 8. Moreover, the restriction ofany of the four to a “cyclotomic line” in the space of characters of Gp∞ – i.e. a coset of the subgroup ofcharacters factoring through the norm map – is a distribution of order 1, and thus not determined uniquelyby its interpolating property. Thus, paradoxically, it is easier to interpolate L-values at a larger set of twiststhan a smaller one.

We know that the module ΛΠ in which our Mazur–Tate elements are valued has rank 1 over OF . We maythus choose an OE -basis ΩΠ of OE ⊗OF ΛΠ . Let us write

µα,α =Lp,(α,α)(Π)

ΩΠ

∈ D(1/2,1/2)(Gp∞ ,E)

and similarly for µα,β , µβ ,α , µβ ,β .

Proposition 8. The distributions

µ+,+ = µα,α +µα,β +µβ ,α +µβ ,β

µ+,− = µα,α −µα,β +µβ ,α −µβ ,β

µ−,+ = µα,α +µα,β −µβ ,α −µβ ,β

µ−,− = µα,α −µα,β −µβ ,α +µβ ,β

all lie in D(1/2,1/2)(Gp∞ ,E), and have the property that if ω is a finite-order character of conductor pnppnp ,with both np,np > 0, then µ∗, vanishes at ω unless ∗= (−1)np and = (−1)np .

Proof. Clear from the interpolating property of the µ’s.

Thus each of the distributions µ∗,, for ∗, ∈ +,−, vanishes at three-quarters of the finite-order char-acters of the p-adic Lie group Gp∞ .

We now interpret this statement in terms of divisibility by half-logarithms. Let log+p be Pollack’s half-logarithm on O×K,p

∼= Z×p (cf. [13]) which is a distribution of order 1/2 with the property that log+p (χ)vanishes at all characters χ of Z×p of conductor pn with n an odd integer (and no others). Similarly, let log−pbe the half-logarithm which vanishes at characters of conductor pn with n positive and even. We define log+pand log−p as distributions on Gp∞ by pushforward, and similarly log±p .

Proposition 9. For ∗, ∈ +,−, the distribution µ∗, is divisible in Dla(Gp∞) by log∗p logp.

Proof. We begin by introducing distributions

µ+,α = µα,α +µβ ,α

µ+,β = µα,β +µβ ,β

µ−,α = µα,α −µβ ,α

µ−,β = µα,β −µβ ,β .

We claim that µ+,α and µ+,β are divisible by log+p , and µ−,α and µ−,β by log−p . We prove only the firstof these four statements, since the proofs of the other three are virtually identical. If χ is any finite-ordercharacter such that np(χ) (the power of p dividing the conductor of χ) is positive and odd, then µ+,α

vanishes at χ by construction.


Since µ+,α is in D(1/2,1/2)(Gp∞ ,E), it follows that it must vanish at any character of G whose pullbackto O×K,p is of finite order with odd p-power conductor. Hence µ+,α is divisible by log+p in D(1/2,1/2)(Gp∞ ,E)(and the quotient clearly has order (0,1/2)).

We now apply the same argument in the other variable, to show that µα,+ = µα,α +µα,β is divisible bylog+p , etc. This shows that µ+,+ is divisible by both log+p and log+p . These two distributions are coprime (ifwe regard them as rigid-analytic functions on the character space of Gp∞ , the intersection of their zero lociis a subvariety of codimension 2) so we are done.

Corollary 2. There exist four bounded measures

L+,+p ,L+,−

p ,L−,+p ,L−,−p ∈ΛF(Gp∞)

such that

Lp,(α,α)(Π)

ΩΠ

=14

(L+,+

p log+p log+p +L+,−p log+p log−p

+L−,+p log−p log+p +L−,−p log−p log−p)

and similarly for the other three unbounded L-functions.

Remark 14. If Π is the base-change to K of a weight 2 modular form, then µα,α and µβ ,β coincide with thedistributions constructed in [9], up to minor differences in the local interpolation factors. Hence the abovecorollary proves the conjecture formulated in op.cit.. However, it does not appear to be possible to constructthe signed distributions L+,+

p etc solely from µα,α and µβ ,β ; it seems to be necessary to use µαβ and µβα

as well, and these last two are apparently not amenable to construction via Rankin–Selberg convolutiontechniques as in op.cit..

References

1. Ash, A., Ginzburg, D.: p-adic L-functions for GL(2n). Invent. Math. 116(1-3), 27–73 (1994).2. Calegari, F., Mazur, B.: Nearly ordinary Galois deformations over arbitrary number fields. J. Inst. Math. Jussieu 8(1),

99–177 (2009).3. Colmez, P.: Fonctions d’une variable p-adique. Astérisque 330, 13–59 (2010).4. Dimitrov, M.: Automorphic symbols, p-adic L-functions and ordinary cohomology of Hilbert modular varieties. Amer. J.

Math. (2013). To appear.5. Haran, S.M.J.: p-adic L-functions for modular forms. Compos. Math. 62(1), 31–46 (1987).6. Harder, G.: Eisenstein cohomology of arithmetic groups. The case GL2. Invent. Math. 89(1), 37–118 (1987).7. Ieso, M.D.: Espaces de fonctions de classe Cr sur OF (2012). Preprint.8. Januszewski, F.: Modular symbols for reductive groups and p-adic Rankin-Selberg convolutions over number fields. J.

Reine Angew. Math. 653, 1–45 (2011).9. Kim, B.D.: Two-variable p-adic L-functions of modular forms for non-ordinary primes (2011). Preprint.

10. Loeffler, D., Zerbes, S.L.: Iwasawa theory and p-adic L-functions for Z2p-extensions (2011). Preprint.

11. Mazur, B., Tate, J.: Refined conjectures of the “Birch and Swinnerton-Dyer type”. Duke Math. J. 54(2), 711–750 (1987).12. Panchishkin, A.A.: Motives over totally real fields and p-adic L-functions. Ann. Inst. Fourier (Grenoble) 44(4), 989–1023

(1994).13. Pollack, R.: On the p-adic L-function of a modular form at a supersingular prime. Duke Math. J. 118(3), 523–558 (2003).14. Washington, L.C.: Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, second edn. Springer-Verlag,

New York (1997)

268 David Loeffler

e-mail: [email protected] L

et p be an odd prime, F/Q an abelian totally real number field, F∞/F its cyclotomic Zp-extension,G∞ = Gal(F∞/Q), A = Zp[[G∞]]. We give an explicit description of the equivariant characteristic idealof H2

Iw(F∞,Zp(m)) over A for all odd m ∈ Z by applying M. Witte’s formulation of an equivariant mainconjecture (or “limit theorem”) due to Burns and Greither. This could shed some light on Greenberg’sconjecture on the vanishing of the λ -invariant of F∞/F.

Key words: cohomology modules, perfect torsion complexes, limit theorem, equivariant characteristicideal.

[email protected]

On equivariant characteristic ideals of real classes

Thong Nguyen Quang Do

0 Introduction

Fix an odd prime p and a Galois extension F/k of totally real number fields, with group G = Gal(F/k).Let F∞ = ∪

n≥0Fn the cyclotomic Zp-extension of F , Γ = Gal(F∞/K), G∞ = Gal(F∞/k), Λ = Zp[[Γ ]],

A = Zp[[G∞]]. Take S f = Sp ∪ Ram(F/k), S = S∞ ∪ S f , where S∞ (resp. Sp) is the set of archimedeanprimes (resp. primes above p) of k and Ram(F/k) is the set of places of k which ramify in F/k. Byabuse of language, we also denote by S the set of primes above S in any extension of k. Let GS(Fn)be the Galois group over Fn of the maximal algebraic S-ramified (i.e. unramified outside S) extension ofFn. Since p is odd and S contains Sp ∪ S∞, it is known that cdp(GS(Fn)) ≤ 2 (see e.g. [NSW08], Pro-pos 8.3.18). Furthermore, the continous cohomology groups H i

S(Fn,Zp(m)) := H i(GS(Fn),Zp(m)), m ∈ Z,coincide with the étale cohomology groups H i

ét(OFn [1/S],Zp(m)). We are interested in the A-modulesH i

Iw,S(F∞,Zp(m)) := lim←−cores

H iS(Fn,Zp(m)) (H i

Iw(.) for short if there is no ambiguity on S), to which can be

attached invariants containing important arithmetical information.For instance, if G is abelian and the µ-invariant associated to F∞/F vanishes, it can be shown that, for

any m≡ 0 (mod 2), the initial A - Fitting ideal of H2Iw(F∞,Zp(m)) is given by the formula :

FitA(H2Iw(F∞,Zp(m))) = twm(IG∞.θS) (1)

where twm denotes the automorphism of the total ring of fractions QA induced by the mth-Iwasawa twistσ 7→ κm

cyc σ−1, κcyc is the cyclotomic character, IG∞ is the augmentation ideal of Zp[[G∞]] and θS ∈ Q A isthe Deligne-Ribet pseudo-measure associated with G∞ ([NQD05], thms 3.1.2, 3.3.3). Note that for m < 0,we implicitly assume the validity of the mth-twist of Leopoldt’s conjecture; see proposition 3.2 below.

The identity (1) was shown by using the (abelian) Equivariant Main Conjecture (EMC for short) ofIwasawa theory in the formulation of Ritter and Weiss ([RW02], theorem 11). Here “equivariant” meansthat the Galois action of G is taken into account. One salient feature of formula (1) is that it allows a proofby descent of the p-part of the Coates-Sinnott conjecture, as well as a weak form of the p-part of Brumer’sconjecture ([NQD05], thms 4.3, 5.2). On the same subject, let us report the approach of Burns and Greither([BG03b], theorems 3.1, 5.1 and corollary 2) using the Iwasawa theory of complexes initiated by Kato andsubsequently extended by Nekovar in [Nek06]. See also [GP13].

After the recent proof of the non commutative EMC (under the hypothesis that µ = 0, see [Kak13],[RW11]), no doubt that the above results could be extended to the case where G is non abelian (this hasbeen done recently by [Nic13a]). But in this article, we are mainly interested in the odd twists, m≡ 1 (mod2), which are not a priori covered by the EMC. Since there are so many “main conjectures” floating around,a short explanation is in order. Let us come back to the classical Main Conjecture, or Wiles’ theorem (WMCfor short, see [Wil90]) and take G = (1) for simplicity. The Galois group X∞ over F∞ of the maximal abelian(p)-ramified pro-p-extension of F∞ is a Λ -torsion module because F is totally real, and the Λ -characteristicseries of X∞ is precisely related by the WMC to the Deligne-Ribet pseudo-measure attached to the maximalpro-p-quotient of Gab

S (F) (where S = S∞ ∪ Sp). If we allow - just for this discussion - the base field F tobe a CM-field, then (torΛX∞)

+ = X+∞ by the weak Leopoldt conjecture, and it is related by Spiegelung

269

270 Thong Nguyen Quang Do

(reflection theorems) to the “minus part” X−∞ of the Galois group over F∞ of the maximal abelian unramifiedpro-p-extension of F∞, so that we know the Λ -characteristic series of X−∞ in terms of Lp-functions. Howeverour intended study of odd twists actually concerns the “plus part” X+

∞ , at least because if ζp ∈ F , thenX ′∞(m−1) is contained in H2

Iw(F∞,Zp(m)) (see propos. 4 below), where X ′∞ is obtained from X∞ by addingthe condition that all (p)-primes (hence all finite primes, because at a non (p)-prime, the local cyclotomicZp-extension coincides with the local maximal non ramified pro-p-extension) must be totally split. A relatedproblem is Greenberg’s celebrated conjecture - a “reasonable” generalization of Vandiver’s - which assertsthe finiteness of X+

∞ (or equivalently of X′+∞ ). Not much is known on the plus parts. Let us recall some results

in the case where k =Q, F is totally real and G is abelian :

• it is well known that the WMC is equivalent to the so-called “Gras type” equality (because it impliesthe Gras conjecture) :

charΛ X∞ = charΛ (U∞/C∞) (2)

where U∞ (resp. C∞) denotes the inverse limit (w.r.t. norms) of the p-completions Un (resp. Cn) ofthe groups of units (resp. circular units) along the cyclotomic tower of F . But the right hand side Λ -characteristic series is not known. Besides, this Gras type equality comes from the multiplicativity ofcharΛ (.) in the exact sequence of class-field theory relative to inertia :

0→U∞/C∞→ U∞/C∞→ X∞→ X∞→ 0,

where U∞ is the semi-local analogue of U∞. But when passing from Λ to A, no straightforward equiv-ariant generalization (of A-characteristic series of modules and their multiplicativity) is known.

• Galois annihilators of X ′∞ have been explicitly computed in [NQDN11] and [Sol10], as well as Fittingideals in the semi-simple case in [NQDN11].

In this context, the main result of this paper will be an explicit equivariant generalization of the Gras typeequality (2) for all odd twists (see thm 2 below).

The proof will proceed in essentially two stages :

• use a “limit theorem” in the style of Burns and Greither ([BG03a], thm 6.1 ; this is also an EMC, but wedon’t call it so for fear of overload), but expressed in the framework of the Iwasawa theory of perfecttorsion complexes as in [Wit06], to relate the A-determinant of H2

Iw(F∞,Zp(m)), m odd, to that of asuitable quotient of H1

Iw(F∞,Zp(m))• use an axiomatization of a method originally introduced by Kraft and Schoof for real quadratic fields

([KS95]) to compute explicitly the latter determinant (which will actually be an A-initial Fitting ideal).

Summary of Iwasawa theoretic notationsF a number fieldF∞ =

⋃n≥0 Fn the cyclotomic Zp-extension of F

Γn = Gal(Fn/F), Γ = Gal(F∞/F), Λ = Zp[[Γ ]]If moreover F/Q is GaloisGn = Gal(Fn/Q), G∞ = Gal(F∞/Q), A= Zp[[G∞]]An (resp. A′n) = the p-group of ideal classes (resp. (p)-classes) of FnX∞ (resp. X ′∞) = lim

←An (resp. A′n) w.r.t. norms

X∞ = the Galois group over F∞ of the maximal abelian pro-p-extension of F∞ which is unramified outsidep

Un = the p-completion of the group of units of FnU∞ = lim

←Un w.r.t. norms

If moreover F/Q is abelian :Cn = the p-completion of the group of circular units of FnC∞ = lim

←Cn w.r.t. norms

On equivariant characteristic ideals of real classes 271

1 Generators and relations, and Fitting ideals

We study in this section an axiomatic method for describing certain (initial) Fitting ideals by generators andrelations. It was first introduced by [KS95] for quadratic real fields, and subsequently applied by [Sol10] tothe cyclotomic field Q(ζp)

+.

1.1 General case

Since the process is purely algebraic, we can relax here the assumptions on k and F imposed in the intro-duction. So F/k will just be a Galois extension with group G, and the usual Galois picture will be :

F∞

k∞

H

Fn

Γ pn

k∞∩F F

Γn

kG0=G

Gn

Note that A= Λ [H] is equal to Λ [G] if and only if k∞∩F = k. Put Rn = Zp[Gn], Rn,a = Z/pa [Gn].Hypothesis : We are given a projective (w.r.t. norms) system of Rn-modules Vn which are Zp-free of finite

type, as well as a projective subsystem Wn ⊂Vn such that each Wn is Rn-free of rank 1. In other words, thereis a norm coherent system η = (ηn), ηn ∈Vn, such that Wn = Rn.ηn for all n≥ 0. Without any originality,a pair (Vn,Wn)n≥0 as above will be called admissible.

At the time being, we don’t worry about the existence of such systems (Wn,Vn)n. Arithmetical exampleswill be given later in sections 2 and 3.

Goal : denoting by Bn the quotient Vn/Wn, describe the module t Bn (where t(.) means Zp-torsion) bygenerators and relations.

In the sequel, for two left modules X ,Y , the Galois action on Hom(X ,Y ) will always be defined byσ f (x) = σ( f (σ−1x)). The Pontryagin dual of X will be denoted by X∗.

Proposition 1. For any n≥ 0,

(t Bn)∗ ' Rn/Dn, where Dn = ∑

σ∈Gn

f (σ−1ηn).σ ; f ∈ Hom(Vn,Zp) .

Proof. We follow essentially the argument of [KS95], thm 2.4. For any a≥ 0, the snake lemma applied tothe pa-th power map yields an exact sequence :

0→ Bn[pa]→Wn/pa→Vn/pa→ Bn/pa→ 0

(the injectivity on the left is due to the fact that Vn is Zp-free).Applying the functor HomGn(. ,Rn,a), we get another exact sequence of Rn,a-modules :

→ HomGn

(Vn,Rn,a

)→ HomGn

(Wn,Rn,a

)→ HomGn

(Bn[pa],Rn,a

)→

But Rn,a is a Gorenstein ring, which means that R∗n,a is a free Rn,a-module of rank 1, hence, for any fi-nite Rn,a-module M, the canonical isomorphism HomRn

(M,R∗n,a

) ∼→M∗ gives rise to an Rn,a-isomorphismHomRn

(M,Rn,a

) ∼→M∗, f 7→ ψ f , where ψ is a chosen Rn,a-generator Rn,a → Qp/Zp. It follows that thefunctor HomRn

(. ,Rn,a

)is exact and we have

→ HomRn

(Vn,Rn,a

)→ HomRn

(Wn,Rn,a

)→ B∗n/pa→ 0


Taking lim←−a

, we derive an exact sequence :

→ HomRn(Vn,Rn)res−→HomRn(Wn,Rn)→ (tBn)

∗→ 0 .

But Wn =Rn.ηn, hence an element f ∈HomRn(Wn,Rn) is determined by the value f (ηn), i.e. HomRn(Wn,Rn)∼→Rn,

f 7→ f (ηn), and the above exact sequence becomes : HomRn(Vn,Rn)→ Rn→ (tBn)∗→ 0.

In other words, (tBn)∗'Rn/ f (ηn) ; f ∈HomGn(Vn,Rn). The canonical isomorphism HomZp(M,Zp)

∼→HomRn(M,Rn),f 7→ F such that F(m) = ∑

σ∈Gn

f (σ−1m).σ , ∀m ∈M, gives the desired result.

Corollary 1. AnnRn

((tBn)

∗)= FitRn

((tBn)

∗)=Dn

Proof. The proposition 1 shows at the same time that (tBn)∗ is Rn-cyclic, hence its Rn-annihilator and

Rn-Fitting ideal coincide, and that FitRn

((tBn)

∗)=Dn.

Definition 1. In the sequel, we shall denote by D(F∞) (or D∞ for short) the projective limit lim←

Dn w.r.t.norms (the detailed transition maps can be found e.g. in [Sol10], proposition 1). Of course D∞ depends onthe admissible pair (Vn,Wn)n≥0. Corollary 1 shows that AnnA(lim← (tBn)

∗) = FitA(lim← (tBn)∗) =D∞.

1.2 A kummerian description

Let E = F(ζp) and ∆ = Gal(E/F). Fix a norm coherent system ζζζ =(ζpn)

n≥0 of generators of the groupsµpn of pn-th roots of unity. Attach to E∞/E the following objects : Un (resp. U ′n) = group of units (resp.(p)-units) of En

Un =Un⊗Zp , U ′n =U ′n⊗Zp

Xn = Galois group over En of the maximal abelian (p)-ramified pro-p-extension of En. At the level of En,we have the Kummer pairing :

H1(GS(En),µpn)×Xn/pn→ µpn , (x,ρ) 7→

(x

1pn)ρ−1

.

Let 〈. , .〉n : Hom(Xn,Z/pn

)×Xn/pn → Z/pn be the pairing defined by

(x

1pn)ρ−1

= ζ〈x,ρ〉npn . Take V E

n to

be Un/tors, or U ′n/tors, or more generally a free Zp-module such that V En /pn → Hom

(GS(En),µpn

). In

this subsection, we relax the condition of Galois freeness on W En , which we assume only to be cyclic :

W En = Z[Jn].ηn, where Jn = Gal(En/k).Define

Dn(E) = ∑σ∈Jn

f (σ−1ηn).σ ; f ∈ (Vn/pn)∗

= ∑σ∈Jn

〈σ−1.ηn,ρ〉nσ ; ρ ∈ Xn/pn

= ∑σ∈Jn

〈ηn,σ .ρ〉n κ−1cyc(σ)σ ; ρ ∈ Xn/pn

(the second equality comes from the Kummer pairing just recalled; the third from a functorial propertyof this pairing). Recall that

Dn(E) := ∑σ∈Jn

f (σ−1ηn).σ ; f ∈ Hom(V E

n ,Zp) .

Proposition 2. With the pair (V En ,W E

n )n≥0 chosen as above

D(E∞) := lim←

Dn(E) = lim←

Dn(E)

is cyclic as a Zp[[Gal(E∞/Q)]]-module.


Proof. Since Hom (V En ,Zp) '

(V E

n ⊗Qp/Zp)∗, the equality lim

←Dn(E) = lim

←Dn(E) is straightforward.

Notice next that in the sum ∑σ∈Jn

〈ηn,σ .ρ〉n κ−1cyc(σ)σ , the value of the pairing 〈ηn,σ .ρ〉n does not depend

on ρ , only on the image of ρ in the cyclic group Gal(

En(η

1pn

n)/En

). Choosing a generator τn of this group,

we see that the sum above is a multiple of ∑σ∈Jn

〈ηn,σ .τn〉n κ−1cyc(σ)σ . By considering the extension of E∞

obtained by adjoining all pn-th roots of all the η ′ns and their conjugates under the action of Gal(E∞/Q), weget the desired cyclicity result.

This proposition will be applied at the end of section 2 by cutting out D(E∞) by characters of Gal(E/F).

As usually happens, the “new” object D∞ appears afterwards to be not so new. Some previous occur-rences must be recorded :

• for k = Q and E = Q(ζp) and for a special choice of ηηη , D∞ plays an important role in Ihara’s theoryof universal power series for Jacobi sums (see [IKY87]). In [NQDN11], § 4.1, D∞ is interpreted asa certain module of “p-adic Gauss sums”, and both [NQDN11] and [Sol10] show that (D∞)

# is theA-initial Fitting ideal of X

′+∞ , where the sign (.)# means inverting Galois action.

• in exactly the same setting, D∞ appears on the Galois side of Sharifi’s conjecture on the two-variableLp-function of Mazur-Kitagawa ([Sha11], propos. 6.2). Note that a weak form of this conjecture hasbeen recently proved for all fields Q(ζpn) by Fukaya and Kato ([FK12]).

• for k totally real and E/k abelian, containing ζp, and for a special choice of ηηη , (D∞)# is shown in

[NQDN11] and [Sol10] to annihilate X′+∞ . Moreover, in the semi-simple case (p |/|G|), thms 3.4.2 and

5.3.2 of [NQDN11] assert that for any odd character ψ of G, the ψ-part ((D∞)#)ψ is isomorphic to

the Fitting ideal of (X ′∞)ψ∗ , where ψ∗ = ωψ−1 denotes the “mirror” of ψ, ω being the Teichmüller

character.• for k/Q abelian and F = k(ζp), D∞ appears prominently in the explicit reciprocity law of Coleman,

generalized by Perrin-Riou ([Fla11] § 1.3; [PR94], thm 4.3.2). Let us just recall the starting point in[PR94] : denoting ⊕

v|pH1(En,v,Zp(m)) by H1

s`(En,Zp(m)) and lim←

H1s`(En,Zp(m)) by H1

Iw,s`(E∞,Zp(m)),

Perrin-Riou constructs a “cup-product” at the infinite level H1Iw,s`(E∞,Zp(1))×H1

Iw,s`(E∞,Zp)∪→(OE ⊗

Zp)[[G∞]] such that, writing πn for the natural projection (OF ⊗Zp) [[G∞]] → Zp[Gn], πn(x∪ y) =∑

σ∈Gn

(σ−1xn∪ yn).σ (recall that the cup-product at finite levels does not commute with corestriction).

2 Semi-simple example

In this section, to illustrate the “Gras type” approach via the WMC, we intend to study a (particular) semi-simple case, which the reader can skip if pressed for time. The following hypotheses will be assumed :Let F/Q be a totally real abelian extension of conductor f , such that p does not divide the order of G =Gal(F/Q). Let UF (resp. U ′F) be the group of units (resp. (p)-units) of F . The group Cyc(F) of F is thesubgroup of F∗ generated by −1 and all the elements NQ(ζ f )/F∩Q(ζ f )(1− ζ a

f ), (a, f ) = 1. The group CF

(resp. C′F ) of circular units (resp. circular (p)-units) in Sinnott’s sense is defined as CF = UF ∩Cyc(F)

(resp. C′F =U ′F ∩Cyc(F)). Write Un, Cn, etc for UFn , CFn , etc and (.) for the p-completion. We take Vn =U ′n(which is Zp-free of rank [Fn : Q]) and look for candidates for the W ′ns. Since p |/|G|, any (p)-place of F istotally ramified in F∞. Let us denote by s the number of (p)-places of F (hence also of F∞). We keep theGalois notations of the beginning of section 1.

Lemma 1. For any n≥ 1, the Zp[Gn]-module C′n is free (necessarily of rank 1) if and only if s = 1.

Proof. Let us first consider only the Λ -module structure. The cohomology groups H i(Γn,C′n) are computed

in general in [NQDL06]. In our situation, C′n is Γn-cohomologically trivial if and only if s = 1 ([NQDL06],propos. 2.8). Since Γn is a p-group, Zp[Γn] is a local algebra, and for a Zp[Γn]-module without torsion,cohomological triviality is equivalent to Zp[Γn]-projectivity, hence to Zp[Γn]-freeness here. To pass from Γn


to Gn, just notice that H i(Gn,C′n)

∆res'H i(Γn,C

′n) because p |/|∆ |, hence s = 1 if and only if C′n is Zp[Gn]-

projective (of Zp-rank equal to |Gn|). By decomposing C′n into χ-parts, C′n = ⊕χ∈G

(C′n)χ , we see that s = 1 if

and only if each (C′n)χ is free over the local algebra Zp[χ][Γn], if and only if C′n is Zp[Gn]-free (necessarily

of rank 1).

Summarizing, if s = 1, one can take Vn =U ′n, Wn =C′n in order to apply proposition 1 to tBn = Bn =U ′n/C′n.Note that in the semi-simple case with s = 1, U ′n/C′n 'Un/Cn for any n ≥ 1 ([NQDL06], lemma 2.7) andalso X ′∞ ' X∞ in the notations of the introduction ([NQDL06], lemma 1.5). Recall that X∞ (resp. X ′∞) is theunramified (resp. totally split at all finite places) Iwasawa module above F∞. We can now determine theA-Fitting ideal of X∞ in our particular case :

Proposition 3. Let F/Q be a totally real abelian extension, such that p |/|G| and s = 1. Then D(F∞)# =

FitA(X∞), where (.)# means inverting the Galois action.

This is a particular case (s = 1) of thm 5.3.2 of [NQDN11]. Typical examples are F = Q(ζp)+, or F =

Q(√

d), d a square free positive integer such that(

dp

)6= 1.

Proof. Since Bn =Un/Cn is finite, proposition 1 shows that D(F∞) =D∞ = FitA(lim←

(B∗n)). It remains to

describe lim←

(B∗n). Denote U∞ = lim←

Un, C∞ = lim←

Cn, Y∞ =U∞/C∞, X0∞ = the maximal finite submodule of

X∞. With s = 1, we have the following codescent exact sequences ([NQDL06], proposition 4.7) :

0−→ (Y∞)Γn −→ Bn −→ (X0∞)

Γn −→ 0 .

Taking duals and lim←

we get an exact sequence of A-modules :

0−→ (X0∞)∗ −→ lim

←(B∗n)−→ α(Y∞)−→ 0, where α(.) denotes the Iwasawa adjoint (with additional ac-

tion by G). In particular, (X0∞)∗ = (lim

←B∗n)

0. Let us take Fitting ideals in this exact sequence. A well knownlemma (generally attributed to Cornacchia ; see e.g. [NQDN11], lemma 3.4.2) states that for any torsionΛ [χ]-module M, FitΛ [χ](M) = FitΛ [χ](M0).charΛ [χ](M) where charΛ [χ](.) denotes the characteristic ideal.In the semi-simple situation, we can put the χ-parts together to get : FitA(lim← B∗n) =FitA(X0

∞).charA(α(Y∞))

(with an obvious definition of charA(.) here). Let (M)# be the module M with inverted Galois ac-tion. It is classically known that α(M) is pseudo-isomorphic to M#, and that FitA(M∗) = FitA(M#)since the p-Sylow subgroups of the G′n’s are cyclic ([MW84], appendix). Hence D∞ = FitA(lim← B∗n) =

FitA(X0∞)

#.charA(Y∞)#. As charA(Y∞) = charA(X∞) in the “Gras type” formulation of the WMC, we get :

(D∞)# = FitA(X0

∞).charA(X∞) = FitA(X∞).

Remarks :

1. In spite of the presence of the algebra A, proposition 3 is not a genuine equivariant result. In particular,the definition of the characteristic ideal charA(.) cannot be generalized to the non semi-simple case.

2. The ideal D(F∞) can easily be made explicit in kummerian terms using § 1.2. It suffices to start from thebase field E = F(ζp) and then use (co) descent from E to F, or from E∞ to F∞, which works smoothlybecause p does not divide [E : F ].

3. It is well known that U ′∞ = H1Iw,Sp

(F∞,Zp(1)) in full generality. In the situation of proposition 3, wehave also X∞ = X ′∞ = H2

Iw,Sp(F∞,Zp(1)) (for details, see proposition 4 below). We can also consider the

modules H2Iw,Sp

(F∞,Zp(m)), m odd, as in the introduction and describe explicitly their A-Fitting idealsby taking Tate twists above E∞, and then doing (co)descent as in 2) (for details, § 3.4.3 below).

3 Equivariant study of the non semi-simple case

In this case, as we noticed before, two major difficulties are encountered right from the start : the notion ofcharacteristic ideals of torsion modules (with appropriate multiplicative properties) is no longer available


; neither is the “Gras type” formulation of the WMC. The solution to both problems will come from theequivariant Iwasawa theory of complexes. Among many existing formulations, that of M. Witte ([Wit06])seems the best suited to our purpose. Let us recall the minimal amount of definitions and results that weneed, referring to [Wit06] for further details or greater generality.

3.1 Perfect torsion complexes and characteristic ideals

(see [Wit06], § 1) Let R be a commutative noetherian ring and Q(R) its total ring of fractions. A complexC· of R-modules is called a torsion complex if Q(R)⊗

RC· is acyclic; perfect if C· is quasi-isomorphic to a

bounded complex of projective R-modules. To a perfect complex C· one can attach its Knudsen-Mumforddeterminant, denoted detR C·. If moreover C· is torsion, the natural isomorphism Q(R) ' Q(R)⊗

RdetR C·

allows to see detR C· as an invertible fractional ideal of R. Recall that these invertible fractional ideals forma group J(R), which is isomorphic to Q(R)×/R× if the ring R is semi-local.

Definition 2. The characteristic ideal of a perfect torsion complex C· is charR(C·) = (detR C·)−1 ∈ J(R).

Examples : If R is a noetherian and normal domain and M is a torsion module which is perfect (consideredas a complex concentrated in degree 0), i.e. of finite projective dimension, then charR(M) coincides withthe “content” of M in the sense of Bourbaki. If R = Λ = Zp[[T ]], then charΛ (M) is the usual characteristicideal. This justifies the name of equivariant characteristic ideal for charA(M).

Many functorial properties of charR(.) are gathered in [Wit06], proposition 1.5. We are particularlyinterested in the following :

If C· is a perfect torsion complex of R-modules such that the cohomology modules of C· are themselvesperfect, then charR(C·) = ∏

n∈Z(charR HnC·)(−1)n

.

3.2 Iwasawa cohomology complexes

For the rest of the paper, unless otherwise specified, F/Q will be an abelian number field (not necessarilytotally real). With the notations and conventions of the beginning of section 1, let us extract from [Wit06], §3 some perfect complexes and cohomology modules for our use (the situation in [Wit06] is more general) :

The Iwasawa complex of Zp(m) relative to S is the cochain complex of continuous étale cohomologyRΓIw(F∞,Zp(m)) as constructed by U. Jannsen. This is a perfect A-complex whose cohomology modulesare H i

Iw(F∞,Zp(m)) = H iIw,S(F∞,Zp(m)) = lim

←cores

H iS(Fn,Zp(m)) for i = 1,2, zero otherwise.

Let us gather in an overall proposition many known properties of these cohomology groups. Our mainreference will be [KNQDF96], sections 1 and 2 (in which S = Sp, but the proofs remain valid for any finiteset S containing Sp). Let us fix again some notations :

E = F(ζp), E∞ = F(ζp∞), Γ× = Gal(E∞/F)r1 (resp. r2) = number of real (resp. complex) places of FUS(E) =US(E)⊗Zp = the p-adic completion of the S-units of EUS(E∞) = lim

←US(En) w.r.t. norms, X ′(E∞) = the totally split Iwasawa module above E∞.

Proposition 4.

(i)For any m ∈ Z, rankZp H1S (F,Zp(m)) = dm +δm, where dm = r1 + r2 (resp. r2) if m is odd (resp. even), and

δm = rankZp(X′(E∞)(m−1)Γ×).

(ii)For any m∈Z, m 6= 0, torZpH1S (F,Zp(m)) = H0

S (F,Qp/Zp(m)). In particular, if m is odd and F totally real,H1

S (F,Zp(m)) is Zp-free.(iii)For any m ∈ Z, m 6= 1, there is a natural codescent exact sequence :

0→(US(E∞)(m−1)/Λ -tors

)Γ×→ H1

S (F,Zp(m))/Zp-tors→ X ′(E∞)(m−1)Γ× → 0


(iv)For any m ∈ Z, m 6= 1, the Poitou-Tate sequence for H2S can be written as : 0→ X ′(E∞)(m− 1)Γ× →

H2S (F,Zp(m))→ ⊕

v∈S f

H2(Fv,Zp(m))' ⊕v∈S f

H0(Fv,Qp/Zp (1−m))∗→ H0S (F,Qp/Zp(1−m))∗→ 0

For m = 1, the leftmost term must be replaced by AS(F), the p-part of the S f -class group of F.

Remark : It is conjectured that X ′(E∞)(m−1)Γ× is finite (i.e. δm = 0) for all m∈Z (see e.g. [KNQDF96],p. 637). These are the so-called mth-twists of Leopoldt’s conjecture. The case m = 0 (resp. 1) corresponds toLeopoldt’s (resp. Gross’) conjecture. For m≥ 2, δm = 0 because K2m−2OF is finite. Recall that we implicitlysuppose that δm = 0 for all m ∈ Z.

At the infinite level, we have the following

Lemma 2. The A-modules H iIw(F∞,Zp(m)), i = 1,2, are perfect.

Proof. Let Q∞ be the cyclotomic Zp-extension of Q and Λ = Zp[[Gal(Q∞/Q)]]. Since Λ is a regularnoetherian ring, every Λ -noetherian module is perfect. For the cohomology modules over A, we just haveto use the following quasi-isomorphism established e.g. by [FK06, Nek06] etc. : the natural ring homomor-phism Λ → A induces a quasi-isomorphism A⊗

ΛR ΓIw(Q∞,Zp(m))−→ R ΓIw(F∞,Zp(m)).

3.3 The limit theorem

As we explained in the introduction, the “limit theorem” of Burns and Greither ([BG03a], thm 6.1) is actu-ally an EMC, expressed in the language of the Iwasawa theory of complexes, which encapsulates equivariantgeneralizations of both the WMC (for the minus part of X∞) and its formulation “à la Gras” (for the pluspart). It is ultimately derived from the WMC, but only after some hard work. We recall here its presentationby M. Witte (only for the characters κm

cyc; the characters in [Wit06] are more general). Our main referencewill be [Wit06], sections 6 and 7. The “limit theorem” will relate a “zeta-element” and a “special cyclotomicelement” constructed as follows :

Let f = N pd , p |/N, the conductor of F . Introduce L∞ = Q(ζN p∞) ⊃ F∞. At the nth-levels, define theStickelberg elements :

Stickn = ∑0<a< f pn(a, f p)=1

( af pn −

12

)(∏

` primeFrob−v`(a)

`

)∈Qp[Gal(Ln/Q)] ,

and at the infinite level : Stick∞ = lim←

Stickn ∈ Q(Zp[[Gal(L∞/Q)]]

).

Denote by pr± the projectors onto the (±1)-eigenspaces of complex conjugation and define the zeta-element LS(F∞,κ

1−mcyc ) as the image by the natural map QZp[[(Gal(L∞/Q))]] −→ QA = QZp[[G∞]] of the

element Twm−1(pr+− pr−Stick∞), where Twa is the Tate twist induced by σ 7→ κacyc(σ).σ .

Example : ([Wit06], proposition 6.3) Let F∞ = Q∞ and S = Sp and write ω for the Teichmüller charac-ter. Then the image by the natural map QZp[[Γ ]]→ Qp of LS(Q∞,κ

1−mcyc ) is Lp(1−m,ωm.rec−1), where

rec : Gal(Q(ζN)/Q)→ (Z/NZ)× is the isomorphism induced by Frob−1` 7→ `, and Lp(.) is the Kubota-

Leopoldt p-adic L-function.Define now the special cyclotomic element ηηη(F∞,κ

mcyc) as the image by the composite map H1

Iw(L∞,Zp(1))=

US(L∞)Twm−1−→ H1

Iw(L∞,Zp(m)) = US(L∞)(m− 1) nat−→H1Iw(F∞,Zp(m)) of the element pr+((1− ζN pn)n≥0).

The element ηηη(F∞,κmcyc) allows to modify the Iwasawa complex RΓIw(F∞,Zp(m)) to get a perfect torsion

complex. Precisely, there exists a unique morphism Aηηη(F∞,κmcyc)[−1]→ RΓIw(F∞,Zp(m)) in the derived

category which induces the natural inclusion on cohomology, hence a unique (up to quasi-isomorphism)complex denoted by RΓIw/ηηη(F∞,κ

mcyc) which takes place in the following distinguished triangle

Aηηη(F∞,κmcyc)[−1]→ R ΓIw(F∞,Zp(m))→ RΓIw/ηηη(F∞,κ

mcyc) .

Lemma 3. ([Wit06], lemma 7.2)

(i)RΓIw/ηηη(F∞,κmcyc) is a perfect torsion complex of A-modules


(ii)If F∞ is totally real and m is odd, then Aηηη(F∞,κmcyc) is a free A-module of rank 1.

We can now state the “limit theorem” ([BG03a], theorem 6.1; [Wit06], theorem 7.4).

Theorem 1. For an abelian number field F/Q, with S = S∞∪Sp∪ Ram(F/Q), for any m ∈ Z :

(i)At any prime P of codimension 1 of A, containing p, the localized complex(RΓIw/ηηη(F∞,κ

mcyc))P

is acyclic(“vanishing of the µ-invariant”).

(ii)LS(F∞,κ1−mcyc ) generates the A-characteristic ideal of RΓIw/ηηη(F∞,κ

mcyc).

To stress the difference between the (equivariant) limit theorem above and results obtained character bycharacter (such as in [HK03]), let us cite the following comparison lemma ([Wit06], lemma 7.6) :

Lemma 4. Suppose for simplification that F is linearly disjoint from Q∞ and p2 - f . Let φ : Ω =

Op[[G∞]]→ Ω = Πχ∈G

Op[[Gal(Q∞/Q)]] be the normalisation of Ω in QΩ , where Op is the ring obtained

from Zp by adding all the values of all the characters of G. Then

charΩ

(Lφ∗ R ΓIw/ηηη(F∞,Op(m)) = LS(F∞,κ

1−mcyc

).Ω

3.4 The determinant of H2Iw(F∞,Zp(m))

From now on, F∞ is totally real and m ∈ Z is odd. For m < 0, we assume implicitly the validity of the“twisted” Leopoldt conjectures (for m > 1, this is a theorem, see proposition 4(i)). To state and prove ourmain result, we shall proceed in several steps.

Because the cohomology A-modules H iIw(.) are perfect (lemma 2), it follows from thm. 1(ii) and the last

property cited in section 3-1 that

charA(H1

Iw(F∞,Zp(m))/Aηηη(F∞,κmcyc))−1

.charA H2Iw(F∞,Zp(m)) = (LS(F∞,κ

1−mcyc )

)= (1) ,

so that it remains only to determine the first equivariant characteristic series, appealing to the algebraicresults of section 1.

3.4.1

The point is to choose the admissible pair (Vn,Wn)n≥0 attached to F∞/F. Fix an odd integer m 6= 1 and forany n ≥ 0, write η

(m)n for the image of ηηη(F∞,κ

mcyc) by the natural map H1

Iw(F∞,Zp(m))→ H1S (Fn,Zp(m)).

Then η(m)n is Zp[Gn]-free : this comes from lemma 3 (ii) and the codescent exact sequence of propos. 4 (i);

for a direct argument, see [BB12], thm. 3.4. Note that highly non trivial ingredients are needed for bothproofs : Bloch-Kato’s reciprocity law for the first, twisted Leopoldt’s conjecture for the second. We shalltake Vn = H1

S (Fn,Zp(m)) and Wn = Zp[Gn].η(m)n . Adding a superscript (.)(m) to the notations of section 1,

let D(m)(F∞) = FitA (lim← B∗n), which we must relate to the desired A-determinant.Remark : The reason for choosing a twist m 6= 1 is that codescent on S-units (corresponding to m = 1)

is notoriously not smooth, especially in case of p-decomposition. We shall reintegrate m = 1 in thm. 2 byusing a “twisting trick”.

Since the p-Sylow subgroups of the Gn’s are no longer necessarily cyclic, the argument on Fitting idealsused in the semi-simple case (proposition 3) no longer works. We must consequently change the definition ofVn = H1

S (Fn,Zp(m)), replacing it by V ′n = image of(H1

Iw(F∞,Zp(m))→ H1S (Fn,Zp(m))

). We don’t change

Wn = Zp[Gn].η(m)n and we define B′n = V ′n/Wn. By proposition 4 (iii), lim

←V ′n = lim

←Vn = H1

Iw(F∞,Zp(m)),

hence lim←

B′n = lim←

Bn := B∞ = H1Iw(F∞,Zp(m))/Aηηη(F∞,κ

mcyc), whereas lim

←B′∗n = α(lim

←B′n) = α(B∞), α(.)

denoting the Iwasawa adjoint, with additional action by H = Gal(F∞/k∞). Note that this adjoint moduleis naturally isomorphic to ExtA(B∞,A) (see e.g. [NQD05], § 3) over A. Over Λ , we know that α(B∞)


and (B∞)# are pseudo-isomorphic, hence the existence of an exact sequence (non canonical) of Λ -modules

0→ α(B∞)→ (B∞)#→Φ → 0, where Φ is a finite abelian p-group. Since H acts on the first two terms, it

also acts on the third, i.e. the above sequence is indeed exact over A.

3.4.2

Since lim←

V ′n = lim←

Vn, the same reasoning exactly as in proposition 1 shows that FitA(lim← B′∗n )=FitA(α(B∞))=

D(m)(F∞) (the same D(m)(F∞) as before). In particular, this Fitting ideal is principal according to proposi-tion 1 and an obvious descent from E∞ = F∞(µp∞) to F∞. It remains to compare the two principal idealsI = FitA(α(B∞)) and J = charA(B∞)

# by using localization :

Lemma 5. ([BG03a], lemma 6.1) Let R be a Cohen-Macaulay ring of dimension 2 and let I,J be twoinvertible fractional ideals of R. Then I = J if and only if IP = JP for all height one prime ideals P of R.

Cutting out if necessary by the characters of the non-p-part of H, we can suppose that our ring A is as inlemma 5 and proceed to localization :

- at a height one prime P not containing p, AP is a discrete valuation ring in which p is invertible. Itfollows that Fit(.) and det(.)−1 coincide over AP, and ΦP = (0), hence IP = JP.

- at the unique height one prime P of A containing p, the vanishing of the µ-invariant (thm 1 (i)) meansthat (B∞)

#P vanishes, hence also α(B∞)P.

We have thus shown that charA(B∞)# = FitA(α(B∞)) =D(m)(F∞). We can now state and prove our main

result :

Theorem 2. Let F/Q be a totally real abelian number field, E = F(ζp), A = Zp[[Gal(F∞/Q)]], B =Zp[[Gal(E∞/Q)]]. Then, for any odd m ∈ Z, we have :

1. charB(H2Iw(E∞,Zp(m)) =D(1)(E∞)

#(m−1) = twm−1D(1)(E∞), where twm−1 is the Iwasawa twist induced

by σ 7→ κm−1cyc (σ)σ−1.

2. charA(H2Iw(F∞,Zp(m)) = (em−1D

(1)(E∞))#, where em−1 is the idempotent of ∆ associated to the power

ωm−1 of a generator ω of ∆ .

Proof. For any odd m 6= 1, we have just seen that

charA(H2Iw(F∞,Zp(m)))# =D(m)(F∞) = νD(m)(E∞) ,

where ν denotes the norm map of ∆ . But D(m)(E∞)=D(1)(E∞)(m−1), so that νD(m)(E∞)= em−1D(1)(E∞),

because ∆ is of order prime to p. Besides, denoting by π the natural projection B→ A, we know thatLπ∗R ΓIw(E∞,Zp(m)) is naturally quasi-isomorphic to R ΓIw(F∞,Zp(m)) ([Wit06], propos. 3.6 (ii)). Hence,by definition of the determinant, charA(H2

Iw(F∞,Zp(m)))= ν (charB H2Iw(E∞,Zp(m)))= em−1(charB H2

Iw(E∞,Zp(1))).Since the idempotent em−1 depends only on the residue class of (m−1) mod (p−1), we can conclude thatcharB H2

Iw(E∞,Zp(1)) = D(1)(E∞)# and charBH2

Iw(E∞,Zp(m)) = D(1)(E∞)#(m− 1) = twm−1D

(1)(E∞) forall odd m ∈ Z.

Note that in the semi-simple case, thm. 2 above contains thm. 5.3.2 of [NQDN11].

3.5 The Fitting ideal of H2Iw(F∞,Zp(m))

The next natural step would be to perform (co)descent on thm. 2. Because cdp GS(F) ≤ 2, the mapH2

Iw(F∞,Zp(m))Γ pn → H2

S (Fn,Zp(m)) is an isomorphism, but the latter module needs no longer be per-fect (or, equivalently, cohomologically trivial) over Zp[Gn]. Actually, in the exact sequence of propos. 4(iv), our knowledge of the cohomology of the codescent module X ′(E∞)(m−1)

Γ pn is . . . less than perfect.A way to turn the difficulty would be to replace determinants by Fitting ideals, which are compatible withcodescent. But for this we need an equivariant analogue of Cornacchia’s lemma which was used in the proofof proposition 3. To this end, we would like to add a technical condition which the reader would rightly findtoo brutal if it were imposed ex abrupto without any explanation. Hence the following preliminaries :


3.5.1

Let us recall briefly the setting of Cornacchia’s lemma : for a noetherian Λ -module M, denote by M0 itsmaximal finite submodule and write M = M/M0. Since the global projective dimension of Λ is equal to 2,pdΛ M ≤ 1 because M0 = (0) (Auslander-Buchsbaum), hence FitΛ (M) = FitΛ (M0).FitΛ (M). If moreoverM is Λ -torsion, FitΛ (M) = charΛ (M) = charΛ (M). The difficulty when passing from Λ to A is that theAuslander-Buchbaum result is no longer available. We shall use instead a weak substitute due to Greither.Recall that A= Λ [H], where H = Gal(F∞/k∞).

Lemma 6. ([Gre00], propos. 2.4) If an A-noetherian torsion module N is cohomologically trivial over Hand has no non-trivial finite submodule, then pdA(N)≤ 1.

In the sequel, we shall work over E∞ = F(ζp∞), fix an odd m ∈ Z, m 6= 1, and consider the module M :=H2

Iw(E∞,Zp(m)). According to proposition 4 and after taking lim←−

with respect to corestriction maps, wehave an exact sequence :

0→ X ′(E∞)(m−1)→M→Wm(E∞)→ 0 ,

where Wm(E∞) is defined tautologically and will be given an explicit description below. This shows inparticular that M0 = X ′(E∞)

0(m−1). We want to get hold of M = M/M0. Applying the snake lemma to thecommutative diagram :

0 // X ′(E∞)(m−1)0 // _

M // M

// 0

0 // X ′(E∞)(m−1) // M // Wm(E∞) // 0 ,

we get an exact sequence 0→ X ′(E∞)(m−1)→ M→Wm(E∞)→ 0. We intend to apply Greither’s lemma tothe rightmost module Wm(E∞), which is the inverse limit of the kernels Wm(En)= Ker

(⊕

v∈S f

H2(En,v,Zp(m))'

⊕v∈S f

H0(En,v,Qp/Zp(1−m))∗→H0(En,Qp/Zp(1−m))∗)

. The limit of the semi-local modules is ⊕v∈S f

H2Iw(E∞,v,Zp(m)),

which is perfect (for the same reason as in lemma 2), hence H-cohomologically trivial. As for the limit ofthe global H0(.)∗’s, note that the transition maps reduce to pa-power maps between twisted roots of unity;since H i(H, lim

←) = lim

←H i(H, .) and H is finite, we readily get the H-cohomological triviality of the limit.

Summarizing, lemma 6 applies and Wm(E∞) has projective dimension ≤ 1. Going down to F∞, we getan exact sequence 0→ em−1X ′(E∞)→ H2

Iw(F∞,Zp(m))→Wm(F∞)→ 0, with pdAWm(F∞) ≤ 1. Since m isodd, the leftmost module lives in the “plus” part. Let E(m−1)

∞ be the subfield of E∞ cut out by the characterωm−1 (notations of thm 2).

Assume Greenberg’s conjecture for E(m−1)∞ /E(m−1), which is equivalent to the vanishing of em−1X ′(E∞)

and yields an exact sequence :

0→ em−1X ′(E∞)0→ H2

Iw(F∞,Zp(m))→Wm(F∞) =˜H2

Iw(F∞,Zp(m))→ 0 (3)

Since pdAWm(F∞) ≤ 1, the Fitting ideal behaves multiplicatively and we have : FitA H2Iw(F∞,Zp(m)) =

FitA(em−1X ′(E∞)

0).FitAWm(F∞). Moreover, FitAWm(F∞)= charAWm(F∞)= charA ˜H2

Iw(F∞,Zp(m))= charA H2Iw(F∞,Zp(m))

(the last equality is obtained by the already used localization argument). Finally :

Theorem 3. For any odd m ∈ Z, m 6= 1, assume Greenberg’s conjecture for E(m−1)∞ /E(m−1). Then :

1. At infinite level,FitA H2

Iw(F∞,Zp(m)) = FitA(em−1 X ′(E∞)0).(em−1D

(1)(E∞))# .

2. At any finite level n≥ 0,

FitZp[Gn] H2S (Fn,Zp(m)) = FitZp[Gn] (em−1 X ′(E∞)

0)Γ pn .(em−1Dn(E))# .


Remark : The perfectness of the two last terms in the exact sequence (3) implies that of the first term, hencethe existence of charA(em−1 X ′(E∞)

0). But the usual localization argument shows that this ideal is (1), andwe recover a particular case of thm. 2 (ii).

3.5.2

Recall that the module Dn(E) was defined in § 1.2 and its quotient mod pn was described explicitly inkummerian terms. The interest of thm. 3 (ii) lies in its comparison with already known results on refine-ments (for m odd) of the Coates-Sinnott conjecture on Galois annihilators of the modules H2

S (Fn,Zp(m))'K2m−2(OFn [1/S], Zp(m)) (by Quillen-Lichtenbaum’s conjecture, now a theorem – more precisely a conse-quence of the so called Milnor-Bloch-Kato conjecture, proved by Voevodsky and others; see e.g. [Wei09]and the references therein). Note that for m odd, the usual formulation of Coates-Sinnott gives no infor-mation other than “zero kills everybody”. A refined conjecture was formulated by [Sna06] (resp. [Nic11])in terms of leading terms (rather than values) of Artin L-functions at negative integers for abelian (resp.general) Galois extensions of number fields, and shown to be a consequence of the equivariant Tamagawanumber conjecture (ETNC) for the Tate motives attached to these extensions. Let us return to the situationof the introduction, where F/k is an abelian extension with group G, k is totally real and F is CM. Weneed to recall quickly the construction of a “canonical fractional ideal” by Snaith and Nickel. We follow thepresentation of [Nic11], adapting it to our situation :

- for the algebraic part, fix m≥ 2 and let H1−m(F) = ⊕S∞

(2πi)m−1Z, with action of complex conjugation

(diagonally on S∞ and on (2πi)m−1). The Borel regulator ρ1−m : K2m−1(OF)→ H+1−m(F)⊗Q yields the

existence of a Q[G]-isomorphism φ1−m : H1−m(F)+⊗Q ∼→K2m−1(OF)⊗Q. This allows, by applying theQuillen-Lichtenbaum conjecture (now a theorem), to construct a G-equivariant embedding (we fix m anddrop the index) φ : H1−m(F)+⊗Zp → H1

S (F,Zp(m)) (note that this H1S (.) does not depend on S⊃ Sp, by

the localization exact sequence in étale cohomology, see [Sou79], III 3). The algebraic part of the canonicalideal will be FitZp[G]((coker φ)∗).

- for the analytic part, consider the ring R(G) of virtual characters of G with values in Q and let L bea finite Galois extension of Q such that each representation of G can be realized over L; let also Vχ be anL[G]-module with character χ . We can form regulator maps :

Rφ1−m : R(G)−→ C∗

χ 7→ det(

ρ1−m.φ1−m | HomG(V∨χ,H+

1−m(F)⊗C)

(where∨χ denotes the contragredient character).

Define then a function ASφ1−m

: R(G)→ C∗, χ 7→ Rφ1−m(χ)/L∗S(1−m,χ), where LS(s,χ) is the S-truncated Artin L-function and L∗S(1−m,χ) is the leading term of this function at (1−m). Gross’ higheranalogue of Stark’s conjecture states that AS

φ1−m(χσ ) = AS

φ1−m(χ)σ for all σ ∈ Aut (C). Assuming this

conjecture and choosing an identification C'Cp, we can now define the “p-adic canonical ideal” (we dropthe index in φ1−m).

Definition 3. KS1−m(p) = (FitZp[G]

(coker φ)∗.(AS

φ)−1))#.

NB : this is actually an “intermediate” ideal on the construction of Snaith-Nickel, but it is all we need.

Theorem 4. ([Sna06, Nic11]) Suppose that Gross’ conjecture, and also the ETNC for the pair (Q(1−m)F ,Z[ 1

2 ][G]), m ≥ 2, hold for the abelian extension F/k, where k is totally real and F is CM. ThenFitZp[G]H2

S (F,Zp(m)) = KS1−m(p).

Proof. See [Nic11], end of the proof of thm 4.1, as well as remark, p. 14.

Remarks and forecasting :

1. For even m ≥ 2, further calculations show that thm 4 contains the p-part of the usual Coates-Sinnottconjecture.


2. If k =Q, Gross’ conjecture and the ETNC hold true, and thm 4 becomes unconditional. Its comparisonwith thm 3 could give an analytic meaning to the parasite modules (em−1X ′(E∞)

0)Γ pn . “Numerical”

information could also be obtained by computing the orders of the groups K2m−2(OFn ,Zp(m)). Forexample, in the semi-simple case, where Γn intervenes in place of Gn, this order was computed by[Mar11] in terms of values of Lp-functions at positive integers.

3. Instead of leading terms of Artin L-functions, one could also appeal to derivatives as in [BdJG12]. Anatural (but resting only on thin air) query would be : is there any conceptual link with thm 2, knowingthat Dn(E) can be interpreted as a module of “p-adic Gauss sums” ([NQDN11], § 4.1) ?

4. A natural expectation would be the extension of thm 2 to the relative abelian case (k 6= Q). But then aserious obstacle is the absence of special elements (at least non conjecturally). Partial progress has beenmade by [Nic13b], starting from the idea of replacing special elements by Lp-functions over k; this isa natural idea since, when k = Q, special elements and Lp-functions are “equivalent” by Coleman’stheory.

5. Finally, one would of course wish to deal with the non abelian case, in view of the non commutativeEMC recently proved by [Kak13] and [RW11]. But a non commutative analogue of the “limit theorem”is missing.

Acknowledgements : It is a pleasure for the author to thank Dr. Malte Witte for many useful discussionson his “limit theorem”, as well as the referee for his careful reading of the manuscript.

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Nearly overconvergent modular forms

Eric Urban

Abstract We introduce and study finite slope nearly overconvergent (elliptic) modular forms. We give anapplication of this notion to the construction of the Rankin-Selberg p-adic L-function on the product of twoeigencurves..

1 Introduction

The purpose of this paper is to define and give the basic properties of nearly overconvergent (elliptic) mod-ular forms. Nearly holomorphic forms were introduced by G. Shimura in the 70’s for proving algebraicityresults for special values of L-functions [20]. He defined the notion of algebraicity of those by evaluatingthem at CM-points. After introducing a sheaf-theoretic definition, it is also possible to give an algebraic andeven integral structure on the space of nearly holomorphic forms, allowing to study congruences betweenthem. This naturally leads to the notion of nearly overconvergent forms. For that matter, we can think thatnearly overconvergent forms are to overconvergent forms what nearly holomorphic forms are to classicalholomorphic modular forms. The notion of nearly overconvergent forms came to the author when workingon his joint project with C. Skinner (see [24] for an account of this work in preparation [22]) and appears asa natural way to study certain p-adic families of nearly holomorphic forms and its application to Bloch-Katotype conjectures. In the aforementioned work where the case of unitary groups1 is considered, the notionis not absolutely necessary but it is clearly in the background of our construction and keeping it in mindmakes the strategy more transparent.

An important feature of nearly overconvergent forms is that its space is equipped with an action of theAtkin operator Up and that this action is completely continuous. This allows to have a spectral decompo-sition and to study p-adic families of nearly overconvergent forms. Another remarkable fact which is notreally surprising but useful is that this space embeds naturally in the space of p-adic forms. In particu-lar, this allows us to define the p-adic q-expansion of these forms. All the tools and differential operatorsthat are used in the classical theory are also available here thanks to the sheaf theoretic definition and theGauss-Manin connection. In particular, we can define the Maass-Shimura differential operator for familiesand the overconvergent projection which is a generalization in this context of the holomorphic projection ofShimura. Our theory is easily generalisable to general Shimura varieties of PEL type. To make this notionmore appealing, I decided to include an illustration (which is not considered in [22]) of its potential appli-cation to the construction of p-adic L-functions in the non-ordinary case. In the works of Hida [12, 13] onthe construction of 3-variable Rankin-Selberg p-adic L-functions attached to ordinary families, the fact thatthe ordinary idempotent is the p-adic equivalent notion of the holomorphic projector makes it play a crucialrole in the construction of Hida’s p-adic measure. Here the spectral theory of the Up-operator on the spaceof nearly overconvergent forms and the overconvergent projection play that important role.

Eric UrbanColumbia University, Department of Mathematics, 2990 Broadway, New York, NY 10027, USA e-mail: [email protected],[email protected]

1 It will be clear to the reader that it could be generalized to any Shimura variety of PEL type.

283

[email protected], [email protected]

[email protected], [email protected]

284 Eric Urban

We now review quickly the content of the different sections. In the section 2, we recall the notion ofnearly holomorphic forms and give its sheaf-theoretic definition. This allows to give an algebraic and in-tegral version for nearly holomorphic forms and define their polynomial q-expansions as well as an arith-metic version of the classical differential operators of this theory. We also check that Shimura’s rationalityof nearly holomorphic forms is equivalent to ours. In section 3, we introduce the space of nearly overcon-vergent forms and we prove it embeds in the space of p-adic forms. We also study the spectral theory ofUp on them and give a q-expansion principle. Then we define the differential operators in families and theoverconvergent projection. In the last section we apply the tools introduced before to make the constructionof the Rankin-Selberg p-adic L-functions on the product of two eigencurve of tame level 1. When restrictedto the ordinary locus, this p-adic L-function is nothing else but the 3-variable p-adic L-function of Hida.

After the basic material of this work was obtained, I learned from M. Harris that he had also given a sheaftheoretic definition2 using the theory of jets which is valid for general Shimura varieties of Shimura’s nearlyholomorphic forms in [9, 10] and the fact his definition is equivalent to Shimura’s has been verified by hisformer student Mark Nappari in his Thesis [17]. It should be easy to see that our description is equivalentto his in the PEL case. However, Harris did not study nor introduce the nearly overconvergent version. Iwould like also to mention that some authors have introduced an ad hoc definition of nearly overconvergentforms as polynomials in E2 with overconvergent forms as coefficients. However this definition cannot begeneralized to other groups and is not convenient for the spectral theory of the Up-operator (or the slopedecomposition). Finally we recently learned from V. Rotger that H. Darmon and V. Rotger have indepen-dently introduced a definition similar to ours in [8] using the work [7].The reader will see that our work isindependent of loc. cit. and recover the result of [7] has a by-product and can therefore be generalized toany Shimura variety of PEL type.

This text grew out from the handwritten lecture notes of a graduate course the author gave at ColumbiaUniversity during the Spring 2012. After, the work [22] was presented at various conferences3 including theIwasawa 2012 conference held in Heidelberg, several colleagues suggested him to write up an account inthe GL(2)-case of this notion of nearly overconvergent forms. This text and in particular the application tothe p-adic Rankin-Selberg L-function would not have existed without their suggestions.

Notations. Throughout this paper p is a fixed prime. Let Q and Qp be, respectively, algebraic closures ofQ and Qp and let C be the field of complex numbers. We fix embeddings ι∞ : Q → C and ιp : Q → Qp.Throughout we implicitly view Q as a subfield of C and Qp via the embeddings ι∞ and ιp. We fix anidentification Qp

∼= C compatible with the embeddings ιp and ι∞. For any rigid analytic space X over ap-adic number field, we denote respectively by A(X),Ab(X),A0(X) and F(X) the rings of analytic function,of bounded analytic function, of analytic functions bounded by 1 and of meromorphic functions on X.

2 Nearly holomorphic modular forms

2.1 Classical definition

In this paragraph, for the purpose to set notations we recall4 some classical definitions and operations onmodular forms.

2.1.1 Definition

We recall that the subgroup of 2× 2 matrices of positive determinant GL2(R)+ acts on on the Poincaréupper half plane

h := τ = x+ iy ∈ C | y > 0

by the usual formula

2 His definition follows a suggestion of P. Deligne.3 The first half of this note was also presented in my lecture given at H. Hida’s 60th birthday conference.4 Those facts are mainly due to Shimura

Nearly overconvergent modular forms 285

γ.τ =aτ +bcτ +d

for γ =(

a bc d

)∈ GL2(R)+ and τ ∈ h

Let f be a complex valued function defined on h. For any integer k ≥ 0 and γ ∈ GL2(R)+, we set

f |kγ(τ) := det(γ)k/2(cτ +d)−k f (γ.τ)

Let r≥ 0 be another integer. We say that f is a nearly holomorphic modular form of weight k and order ≤ rfor an arithmetic group Γ ⊂ SL2(R) if f satisfies the following properties:

(a) f is C∞ on h,(b) f |kγ = f for all γ ∈ Γ ,(c)There are holomorphic functions f0, . . . , fr on h such that

f (τ) = f0(τ)+1y

f1(τ)+ · · ·+1yr fr(τ)

(d) f has a finite limit at the cusps.

If f is a C∞ function on h, we set following Shimura’s ε. f the function of h defined by

(ε. f )(τ) := 8πiy2 ∂ f∂ τ

(τ) (1)

It is easy to check that the condition (c) can be replaced by the condition

(c’)εr+1. f = 0

Moreover if f is nearly holomorphic of weight k and order ≤ r, then ε. f is nearly holomorphic of weightk− 2 and order ≤ r− 1. We denote by Nr

k(Γ ,C) the space of nearly holomorphic form of weight k, order≤ r and level Γ . For r = 0, this is the space of holomorphic modular form of weight k and level Γ and wewill sometimes use the standard notation Mk(Γ ,C) instead of N0

k(Γ ,C).

2.1.2 The Maass-Shimura differential operator

We recall the differential operator δk on the space of nearly holomorphic forms of weight k defined by

δk. f :=1

2iπy−k ∂

∂τ(yk f ) =

12iπ

(∂ f∂τ

+k

2iyf )

An easy computation shows that δk. f is of weight k+2 and its degree of near holomorphy is increased byone. For a positive integer s, let δ s

k be the differential operator defined by

δsk := δk+2s−2 · · · δk

The following easy lemma is due to Shimura and is proved by induction on r. We will generalize it to nearlyoverconvergent forms in the next section.

Lemma 1. Let f ∈ Nrk(Γ ,C). Assume that k > 2r. Then there exist g0, . . . ,gr with gi ∈Mk−2i(Γ ,C) such

thatf = g0 +δk−2g1 + · · ·+δ

rk−2rgr

It is easy to check, the forms gi’s are unique. When the hypothesis of this lemma are satisfied, Shimuradefines the holomorphic projection H( f ) of f as the holomorphic form defined by

H( f ) := g0

Remark 1. The conclusion of this lemma is wrong if the assumption k > 2r is not satisfied. The mostimportant example is given by the Eisenstein series E2 of weight 2 and level 1.

286 Eric Urban

2.2 Sheaf theoretic definition

2.2.1 Sheaves of differential forms

Let Y = YΓ := Γ \H and X = XΓ := Γ \(htP1(Q)) be respectively the open modular curve and completemodular curve of level Γ . Let E be the universal elliptic curve over YΓ and let p : E→ XΓ be its Kuga-Satocompactification over XΓ . We consider the sheaf of invariant relative differential forms with logarithmicpoles along ∂ E= E\E which is a normal crossing divisor of E.

ω := p∗Ω 1E/X (log(∂ E))

It is a locally free sheaf of rank one in the holomorphic topos of X . We also consider

H1dR := R1p∗Ω •E/X (log(∂ E))

the sheaf of relative degree one de Rham cohomology of E over X with logarithmic poles along ∂ E. TheHodge filtration induces the exact sequence

0→ ω →H1dR→ ω

∨→ 0 (1)

and in the C∞-topos, this exact sequence splits to give the Hodge decomposition:

H1dR = ω⊕ω

2.2.2 Complex uniformization

More concretely, let π be the projection h→ YΓ and π∗E be the pull back of E by π . We have

π∗E= (C×h)/Z2

where the action of Z2 on C×h is defines by (z,τ).(a,b) = (z+a+bτ,τ) for (z,τ)∈C×h and (a,b)∈Z2.The fiber Eτ of π∗E at τ ∈ h can be identified with C/Lτ with Lτ = Z+ τZ⊂ C. We have

π∗ω = Ohdz

with Oh the sheaf of holomorphic function on h. Note also that

E= Γ \C×h/Z2

with the action of Γ on C×h/Z2 given by γ.(z,τ) = ((cτ +d)−1z,γ.τ). We therefore have

γ∗dz = (cτ +d)−1dz

From this relation and the condition at the cusps, it is easy and well known to see that

H0(XΓ ,ω⊗k)∼=Mk(Γ ,C)

Let C∞h the sheaf of C∞ functions on h. Then the Hodge decomposition of π∗H1

dR reads

π∗H1

dR⊗C∞h = C∞

hdz⊕C∞hdz

On the other hand, by the Riemann-Hilbert correspondence, we have

π∗H1

dR = π∗R1p∗Z⊗Oh = Hom(R1p∗Z,Oh) = Ohα⊕Ohβ

where α,β is the basis of horizontal sections inducing on H1(Eτ ,Z) = Lτ the linear forms α(a+ bτ) = aand β (a+bτ) = b so that we have


dz = α + τβ and dz = α + τβ

From the action of Γ on the differential form dz, it is then easy to see that

γ∗ ·(

dzβ

)=

((cτ +d)−1

−c0

(cτ +d)

)(dzβ

)We define the holomorphic sheaf of XΓ :

Hrk := ω

⊗k−r ⊗Symr(H1dR)

Then we have the following proposition.

Proposition 1. The Hodge decomposition induces a canonical isomorphism

H0(XΓ ,Hrk)∼=Nr

k(Γ ,C)

Proof. Let η ∈H0(XΓ ,Hrk). Then π∗η(τ) =∑

rl=0 fl(τ)dz⊗

k−lβ⊗

lwhere the fl’s are holomorphic functions

on h. Since we have β = 12iy (dz−dz), we deduce:

π∗η(τ) =

r

∑l=0

fl(τ)

(2iy)l

l

∑i=0

(−1)i(

li

)dz⊗

k−idz⊗

i

The projection of π∗η on the (k,0)-component of H0(h,π∗Hrk) is therefore given by f (τ)dz⊗

kwith

f (τ) =r

∑l=0

fl(τ)

(2iy)l

It is clearly a nearly holomorphic form. It is useful to remark that the projection on the (k,0)-componentis injective5. Conversely, if f (τ) = ∑

rl=0

fl(τ)yl is a nearly holomorphic form of weight k and order ≤

r, using the injectivity of the projection onto the (k,0)-component, it is straightforward to see that∑

rl=0(2i)l fl(τ)dz⊗

k−lβ⊗

lis invariant by Γ and defines an element of H0(XΓ ,H

rk) projecting onto f (τ)dz⊗

k

via the Hodge decomposition. utThe quotient by the first step of the de Rham filtration of Hr

k induces by Poincaré duality the followingcanonical exact sequence

0→ ω⊗k →Hr

k→Hr−1k−2→ 0 (2)

The map Hrk→Hr−1

k−2 induces the morphism ε of (1).

2.3 Rational and integral structures

2.3.1 Rational and integral nearly holomorphic forms

Let N be a positive integer and let us assume Γ =Γ1(N) with N ≥ 3. Then XΓ = X1(N) is defined over Z[ 1N ]

as well as ω , H1dR and Hr

k. Recall that YΓ = Y1(N) classifies the isomorphism classes of pairs (E,αN)/S

where E/S is an elliptic scheme over an Z[ 1N ]-schemes S and αN is a Γ1(N)-level structure for E (i. e. an

injection of group scheme: µN/S → E[N]/S). Moreover the generalized universal elliptic curve is definedover X1(N) and we can define the sheaves ω , H1

dR and Hrk over X1(N)/Z[ 1

N ] as in the previous paragraph.

The exact seqeunce (2) is also defined over Q. For any Z[ 1N ]-algebra A, we define Mk(N,A) and Nr

k(N,A)respectively as the global section of ω

⊗k/A and Hr

k/A. This gives integral and rational definitions of the spaceof nearly holomorphic modular forms.

5 We will see a similar fact in the p-adic case. See Proposition 6.

288 Eric Urban

2.3.2 Nearly holomorphic forms as functors

It follows from the exact sequence (1) that H1dR is locally free over Y1(N)Z[1/N]. A similar result would hold

for general Shimura varieties of PEL type. In that case the torsion-freeness result from the basic propertiesof relative de Rham cohomology (for example see [15]). We may therefore consider T the B-torsor over(Y1(N))Zar of isomorphisms ψU : H1

dR/U∼= OU ⊕OU inducing an isomorphism ψ1

U : ω/U∼= OU ⊕0 such

that the isomorphism (H1dR/ω)/U

∼= OU induced by ψU is dual to ψ1U for all Zariski open subset U ⊂ YΓ .

For any ring R, we denote by R[X ]r the R-module of polynomial in X of degree ≤ r. Let B the Borelsubgroup of SL2 of upper triangular matrices. Then we consider the representation ρr

k of B(R) on R[X ]rdefined by

ρrk((

a b0 a−1

)).P(X) = akP(a−2X +ba−1)

Then over Y1(N), for any Z[ 1N ]-algebra, we have

T×B A[X ]r(k)∼=Hrk/A

This can be checked easily for principal rings and this implies the isomorphism above since the formationof both left and right hand sides commute to base change.

Concretely, we can see it the following way. For an elliptic scheme E/R we consider a basis (ω,ω ′) ofH1

dR(E/R) such that ω is a basis of ωE/R = H0(E,Ω 1E/R) and 〈ω,ω ′〉dR = 1. Then f ∈ Nr

k(N,A) can beseen as a functorial rule assigning to any A-algebra R and a quadruplet (E,αN ,ω,ω ′)/R a polynomial

f (E,αN ,ω,ω ′)(X) =r

∑l=0

blX l ∈ R[X ]r

defined such that the pull-back of f to H1dR(E/R) is ∑

rl=0 blω

⊗k−l ⊗ω ′⊗l. For any a ∈ R×,b ∈ R, we have

f (E,αN ,aω,a−1ω′+bω)(X) = a−k f (E,αN ,ω,ω ′)(a2X−ab)

The condition that f is finite at the cusps is expressed in terms of q-expansion as usual. It will be defined inthe next paragraph.

Proposition 2. Let f ∈Nrk(N,A) and ε. f ∈Nr−1

k−2(N,A) the image of f by the projection (2). Then for anyquadruplet (E,αN ,ω,ω ′)/R, we have

(ε. f )(E,αN ,ω,ω ′)(X) =d

dXf (E,αN ,ω,ω ′)(X)

Proof. For any ring R, we have the exact sequence:

0→ R[X ]0(k)→ R[X ]r(k)→ R[X ]r−1(k−2)

where the right hand side map is given by P(X) 7→ P′(X). It is clearly B(R)-equivariant. By taking thecontracted product of this exact sequence with T, we obtain the exact sequence of sheaves (2) which impliesour claim. Notice that the map of sheaves inducing ε is surjective only if r! is invertible in A. ut

2.3.3 Polynomial q-expansions

We consider Tate(q) the Tate curve over Z[ 1N ]((q)) with its canonical invariant differential form ωcan and

canonical Γ1(N) level structure αN,can. We have the Gauss-Manin connection:

∇ : H1dR(Tate(q)/Z[

1N]((q)))→H1

dR(Tate(q)/Z[1N]((q)))⊗Ω

1Z[ 1

N ]((q)))/Z[ 1N ]

and letucan := ∇(q

ddq

)(ωcan)


Then (ωcan,ucan) form a basis of H1dR(Tate(q)/Z[ 1

N ]((q))) and ucan is horizontal; moreover 〈ωcan,ucan〉dR =

1 (see for instance the appendix 2 of [16]). For any Z[ 1N ]-algebra A and f ∈Nr

k(N,A), we consider

f (q,X) := f (Tate(q)/A((q)),αcan,ωcan,ucan)(X) ∈ A[[q]][X ]r.

We call it the polynomial q-expansion of the nearly holomorphic form f .

Remark 2. We can think of the variable X as −14πy .

2.3.4 The nearly holomorphic form E2

It is well-known that the Eisenstein series of weight 2 and level 1 is a nearly holomorphic form of order 1.Its given by

E2(τ) =−1

24+

18πy

+∞

∑n=1

σ1n qn

where σ1n is the sum of the positive divisor of n and q = e2iπτ .

We can define E2 as a functorial rule. Let R be a ring with 16 ∈ R and E be an elliptic curve over R. Recall

(see for instance [16, Appendix 1]) that any basis ω ∈ ωE/R defines a Weierstrass equation for E:

Y 2 = 4X3−g2X−g3

such that ω = dXY and η =X dX

Y form a R-basis of H1dR(E/R). Moreover if we replace ω by λω , η is replaced

by λ−1η . Therefore ω ⊗η ∈ ωE/R⊗H1dR(E/R) is independant of the choice of ω . It therefore defines a

section of H12 over Y/Z[ 1

6 ]. If (ω,ω ′) is a basis of H1

dR(E/R) such that 〈ω,ω ′〉dR = 1 then we can put

E2(E,ω,ω ′) = 〈η ,ω ′〉dR +X〈ω,η〉dR

where 〈·, ·〉dR stands for the Poincaré pairing on H1dR(E/R).

Its polynomial q-expansion is given by

E2(q,X) = E2(Tate(q),ωcan,ucan)(X) =P(q)12

+X

because ucan =−P(q)12 ωcan +ηcan and 〈ωcan,ηcan〉dR = 1 and where P(q) is defined in [16] by

P(q) = 1−24∞

∑n=1

σ1n qn

Since the q-expansion of E2 is finite at q = 0, E2 defines a section of H12 over X . From the q-expansion, it is

easy to see that E2 =−2E2. We clearly have ε.E2 = 1 (i.e. is the constant modular form of weight 0 takingthe value 1).

Remark 3. We can see easily that multiplying by E2 is useful to get a splitting of H1k :

0→ ω⊗k →H1

k → ω⊗k−2 → 0

In particular, that shows that nearly holomorphic forms are polynomial in E2 with holomorphic forms ascoefficients. As mentioned in the introduction, this provides a way to give an ad hoc definition of nearlyoverconvergent forms.

2.4 Differential operators

Recall that the Gauss-Manin connection

290 Eric Urban

∇ : H1dR→H1

dR⊗ΩX1(N)/Z[ 1N ](log(Cusp))

induces the Kodaira-Spence isomorphism ω⊗2 ∼= ΩX1(N)/Z[ 1

N ](log(Cusp)) and a connection

∇ : Symk(H1dR)→ Symk(H1

dR)⊗ΩX1(N)/Z[ 1N ](log(Cusp))∼= Symk(H1

dR)⊗ω⊗2

The Hodge filtration of Symk(H1dR) is given by Fil0 ⊃ ·· · ⊃ Filk ⊃ Filk+1 = 0 with

Filk−r =Hrk = ω

⊗k−r ⊗Symr(H1dR) for 0≤ r ≤ k

Since ∇ satisfies Griffith transversality, when k≥ r≥ 0, it sends Filk−r into Filk−r−1⊗ΩX1(N)/Z[ 1N ](log(Cusp)).

We therefore get the sheaf theoretic version of the Maass-Shimura operator6:

δk : Hrk→Hr+1

k+2 for 0≤ r ≤ k

We still denote δk the corresponding operator

δk : Nrk(N,A)→Nr+1

k+2(N,A)

The following proposition gives the effect of δk on the polynomial q-expansion.

Proposition 3. Let f ∈Nrk(N,A). Then the polynomial q-expansion of δk f is given by:

(δk f )(q,X) = XkD(X−k f (q,X))

where D is the differential operator on A[[q]][X ] given by D = q ∂

∂q −X2 ∂

∂X . In other words, if f (q,X) =

∑ri=0 fi(q)X i, we have

(δk f )(q,X) =r

∑i=0

qddq

fi(q)X i +(k− i) fi(q)X i+1

Proof. The pull-back of f to φ ∗Hrk with φ : Spec(A((q)))→X1(N)/A corresponds to (Tate(q),αN,can)/A((q))

is given by

φ∗ f =

r

∑i=0

fi(q)ω⊗k−i

can ⊗u⊗i

can

Since ∇ induces dqq∼= ω2

can, ∇(q ddq )(ωcan) = ucan and ∇(q d

dq )(ucan) = 0„ we have:

∇(qd

dq)(φ ∗ f ) =

r

∑i=0

ddq

fi(q)ω⊗k−i+2

can ⊗u⊗i

can +(k− i) fi(q)ω⊗k−i−1

can ⊗u⊗i+1

can

This implies our claim by the definition of the polynomial q-expansion. ut

Remark 4. We can rewrite the formula for δk in the following way:

δk = D+ kX

Using the relation [D,X ] =−X2, we easily show by induction that

δrk =

r

∑j=0

(rj

)Γ (k+ r)

Γ (k+ r− j).X jDr− j (1)

Notice in particular that for s≤ r and h holomorphic, we have

(εsk+2r δ

rk )h =

r

∑j=s

(rj

)Γ (k+ r)Γ ( j+1)

Γ (k+ r− j)Γ ( j+1− s).X j−s(q

ddq

)r− jh (2)

6 We leave it as an exercise to check that this operator corresponds to the classical Maass-Shimura operator via the isomorphismof Proposition 1.


2.5 Hecke operators

Let R1(N) be the abstract Hecke algebra attached the pair (Γ1(N),∆1(N)) by Shimura [21, chapt. 3]. Thisalgebra is generated over Z by the operators Tn for n running in the set of natural integers. If ` is a primedividng N, the operator T` is sometimes called U`. These operators act on the space of nearly holomorphicforms by the usual standard formulas and preserve the weight and degree of nearly holomorphy. Moreoverthe Hecke operators respect the rationality. After an appropriate normalization, it can be seen that integralityis preserved as well.

For any ring A⊂ C, we then denote by hrk(N,A)⊂ EndC(N

rk(N,C)) the subalgebra generated over A by

the image of the Tn’s. If Z[ 1N ]⊂ A⊂ B, then the above remark shows that we have

hrk(N,A)⊗A B∼= hr

k(N,B)

An easy computation shows that, for each integer n and f ∈Nrk(N,C) we have:

(δk. f )|k+2Tn = n.δk( f |kTn)

ε.( f |kTn) = n.(ε. f )|k−2Tn

Remark 5. From these formulae, we see that if f is a holomorphic eigenform of weight k. Then δ rk f is

a nearly holomorphic eigenform of weight k + 2r. Moreover the system of Hecke eigenvalues of δ rk f is

different than the one of any holomorphic Hecke eigenform of weight k+2r if r > 0.

2.6 Rationality and CM-points

2.6.1 Evaluation at CM-points

We review quickly the rationality notion introduced by Shimura. For K ⊂ Q an imaginary quadratic fieldand τ ∈ h∩K, the elliptic curve Eτ has complex multiplication by K and is therefore defined over Kab ⊂Qthe maximal abelian extension of K by the theory of Complex Multiplication. We then denote by ωτ aninvariant Kähler differential of Eτ defined over Kab and we denote by Ωτ the corresponding CM perioddefined by

ωτ = Ωτ dz

Then (ωτ ,ωτ) forms a basis of H1dR(Eτ/K

ab). Let E be a number field and f ∈ Nrk(N,E). Let αN,τ the

Γ1(N)-level structure of Eτ induced by 1N Z/Z⊂C/Lτ . Then the polynomial f (Eτ ,αN,τ ,ωτ ,ωτ)(X) belongs

to EKab[X ]. Since we have ωτ = Ω τ dz, we deduce that f (Eτ ,αN,τ ,ωτ ,ωτ)(0) is the left hand side of (1)and that we therefore have

f (τ)Ω k

τ

∈ EKab (1)

According to Shimura, a nearly holomorphic form is defined as rational if and only if it satisfies (1) for anyimaginary quadratic field K and almost all τ ∈ h∩K. It can be easily seen his definition is equivalent to oursheaf theoretic definition.

Proposition 4. Let f ∈ Nrk(N,C) and E be a number field such that for any imaginary quadratic field

K⊂Q and almost all τ ∈ h∩K, we have

f (τ)Ω k

τ

∈ E.Kab,

then, f ∈Nrk(N,EQab).

Proof. We just give a sketch under the assumption k > 2r since the general case can be deduced aftermultiplying f by E2. Thanks to a Galois descent argument, we may assume E contains the eigenvalues ofall Hecke operators acting on Nr

k(N,C). By Lemma 1, we can decompose f as

292 Eric Urban

f = f0 +δk−2 f1 + · · ·+δrk−2r fr

with fi holomorphic of weight k− 2i. Now we remark that if T is a Hecke operator defined over E, thenf |kT satisfies (1). This follows easily from the definition of the action of Hecke operators using isogenies.Moreover from Remark 5, the system of Hecke eigenvalues of nearly holomorphic forms δ ih and δ i′h′ aredistinct for any two holomorphic forms h and h′ when i 6= i′; we deduce that δ i fi satisfy (1). We may assumetherefore f = δ r

k−2rg for an holomorphic form g of weight k−2r. In fact using a similar argument, we mayeven assume g is an eigenform. Then g = λ .g0 with g0 defined over E. Since δ r

k−2rg0 is defined over E, wededuce from §2.6.1 that δ r

k−2rg0 satisfies (1) and therefore λ ∈ EKab and f ∈Nrk(N,EKab). Since this can

be done for any K the result follows. ut

3 Nearly overconvergent forms

In this section, we introduce our definition of nearly overconvergent modular forms and show they are p-adic modular forms of a special type. We use the spectral theory of the Atkin Up-operator on them and wedefine p-adic families of such forms. We study also the effect of differential operators on them and define ananalogue of the holomorphic projection. These tools are useful to study certain p-adic families of modularforms and also to study p-adic L-functions.

3.1 Katz p-adic modular forms

3.1.1 Definition

We fix p a prime. Let Xrig be the generic fiber in the sense of rigid geometry of the formal completion ofX/Zp along its special fiber. Let A ∈ H0(X/Fp ,ω

⊗p−1) be the Hasse invariant and let Aq be a lifting of Aq

to characteristic 0 for q sufficiently large. For ρ ∈ pQ ∩ [p−1/p+1,1], we write X≥ρ for the rigid affinoidsubspace of Xrig defined as the set of x ∈ Xrig satisfying |Aq(x)|p ≥ ρq. For ρ = 1 we get the ordinary locusof Xrig and we denote it Xord. The space of p-adic modular forms of weight k is defined as

Mp−adick (N) := H0(Xord,ω

⊗k)

The space of overconvergent forms of weight k is the subspace of p-adic forms which are defined on somestrict neighborhood of Xord so:

M†k (N) := lim→

ρ<1

H0(X≥ρ ,ω⊗k)

3.1.2 Frobenius and Θ

Let ϕ : Xord → Xord the lifting of Frobenius induced on Yord by (E,αN) 7→ (E(ϕ),α(ϕ)N ) where E(ϕ) :=

E/E[p] and α(ϕ)N is the composition of αN and the Frobenius isogeny E → E(ϕ). We get a ϕ∗-linear

morphism obtained as the composite

Φ : H1dR→ ϕ

∗H1dR =H1

dR(Eϕ/Xord)→H1

dR

This morphism stabilizes the Hodge filtration of H1dR and we know by Dwork that there is a unique Φ-stable

splitting, called the unit root splitting:

H1dR/Xord = ω/Xord ⊕U/Xord

such that Φ is invertible on U and U is a free sheaf of rank 1 generated by its sub-sheaf of horizontal sectionsfor the Gauss-Manin connection. This unit root splitting induces a splitting of Hr

k/Xord and therefore of a


canonical projectionHr

k/Xord → ω⊗k

/Xord(1)

We now recall the definition of the Theta operator on the space of p-adic modular forms of weight k. At thelevel of sheaves, Θ is defined as the composite of the following maps of sheaves over the ordinary locus:

ω⊗k

/Xord

δk→H1k+2/Xord → ω

⊗k+2

/Xord

where he second arrow is the one given by (1) for r = 1. This defines

Θ : Mp−adick (N)→Mp−adic

k+2 (N)

It follows from the Proposition 3 and Proposition 6 below that on the level of q-expansion, we have:

Θ( f )(q) = qd

dqf (q)

for all f ∈Mp−adick (N). The following proposition will be useful in the next paragraph.

Proposition 5. For any ρ < 1 and any Zariski open V ⊂ X≥ρ , the unit root splitting on Vord := U ∩Xorddoes not extend to a splitting of the Hodge filtration of H1

dR over any finite cover of V .

Proof. We show it by contradiction. Let us assume that this splitting extends to some finite cover S of Vfor some ρ < 1.

H1dR(E/S) = ωE/S⊕U(E/S).

Let Sord = S×Vord V . Since U(E/S)⊗OSord is stable by Φ so is U(E/S). Since V is a strict neighborhoodof Vord, we can find a finite extension L of Qp and x ∈ S(L)\Sord(L). Then, we will obtain a splittingH1

dR(Ex/L) = ωEx/L⊕U(Ex/L) with Φx inducing a semi-linear invertible endomorphism of U(Ex/L). LetkL be the residue field of L. By the results of [3, chapt. 7], the pair (H1

dR(Ex/L),Φx) is isomorphic to(H1

crys(Ex,0/W (kL))⊗ L,F∗⊗ idL) where H1crys(Ex,0/W (kL)) stands for the crystalline cohomology of the

special fiber Ex,0/kL of Ex over the ring of Witt vectors of kL and where Φx = F∗⊗ idL where F∗ is the

“ crystalline " Frobenius induced by the Frobenius isogeny Ex,0 → E(p)x,0 in characteristic p. Since it has a

splitting of the form H1crys(Ex,0/OL) = Fil1⊕U with F∗ inversible on U , Ex,0 has to be ordinary which is a

contradiction since x /∈ Sord(L). ut

3.2 Nearly overconvergent forms as p-adic modular forms

3.2.1 Definition of nearly overconvergent forms

For each ρ , H0(X≥ρ ,Hrk) is naturally a Qp-Banach space for the Supremum norm | · |ρ and if ρ ′ < ρ < 1,

the map H0(X≥ρ ′ ,Hrk) → H0(X≥ρ ,Hr

k) is completely continuous. We define the space of nearly overcon-vergent forms of weight k and order ≤ r by

Nr,†k (N) := lim→

ρ<1

Nr,ρk (N)

with Nr,ρk (N) := H0(X≥ρ ,Hr

k). We can define the operators δk and ε on nearly overconvergent forms sincethey are defined at the level of sheaves. Moreover we can define the polynomial q-expansion of a nearlyoverconvergent form and it is straightforward to check that the action of δk and ε on this q-expansion is thesame as for nearly holomorphic forms.

Remark 6. For any nearly overconvergent form f of weight k and order at most r, we can easily show thatthere exist overconvergent forms g0, . . . ,gr such that

f = g0 +g1.E2 + · · ·+gr.Er2

294 Eric Urban

where for i = 0, . . . ,r, gi is of weight k−2i. This could be used as an ad hoc definition of nearly overconver-gent forms but it would be uneasy to show this space is stable by the action of Hecke operators and that ithas a slope decomposition as we will see in the next sections. Moreover, this definition would not be suitedfor generalization to higher rank reductive groups.

3.2.2 Embedding into the space of p-adic modular forms

We now want to consider nearly overconvergent modular forms as p-adic modular forms. Using (1), we geta map

H0(X≥ρ ,Hrk)→ H0(Xord,H

rk)→ H0(Xord,ω

⊗k) (1)

We have the following

Proposition 6. The maps (1) induces a canonical injection

Nr,†k (N) →Mp−adic

k (N) (2)

fitting in the commutative diagram:

Nr,†k (N) _

// Mp−adick (N) _

Qp[[q]][X ]r // Qp[[q]]

where the bottom map is induced by evaluating X = 0.

Proof. The fact that the diagram commutes follows from the fact that ucan belongs to the fiber of the unitroot sheaf U at the Zp((q))-point defining Tate(q). Indeed, it is explained in the appendix 2 of [16] thatucan is fixed by Frobenius. We are left with proving the injectivity. We consider f in the kernel of this map.Let U ⊂ X≥r ∩Yrig be an irreducible affinoid. It is the generic fiber in the sense of Raynaud of an affineformal scheme Sp f (R) with R a p-adically complete domain. Let E/R the universal elliptic curve over R.Let us choose a basis (ω,ω ′) of H1

dR(E/R) as in the previous section and such that 〈ω,ω ′〉dR = 1. Let h∈ R

be a lifting of the Hasse invariant of E×R Spec(R/pR) and let S := R[1/h] where the hat here stands forp-adic completion. Then E/S has ordinary reduction and the unit root splitting over S defines a basis (ω,u)of H1

dR(E/S). We must haveu = ω

′+λ .ω

with λ ∈ S. But U is a strict neighborhood of Uord =U ∩Xord which is the generic fiber of Sp f (S), we knowby the previous proposition that λ is not algebraic over R. Let Q(X) := f (E/R,αN ,ω,ω ′)(X) ∈ R[X ]r.We want to show that Q(X) = 0. By assumption, we know that Q(λ ) = f (E/S,αN ,ω,ω ′ + λω)(0) =f (E/S,αN ,ω,u)(0) = 0. Since λ is not algebraic over R, this is possible only if Q(X) = 0. Since this can bedone for any pair (ω,ω ′) we conclude that f ≡ 0. ut

If f is a nearly overconvergent form, the p-adic q-expansion of f is by definition the q-expansion ofthe image of f in the space of p-adic forms. The following corollary can be thought of as a polynomialq-expansion principle for the degree of near overconvergence.

Corollary 1. Let f ∈ Nr,†k (N). If f (q,X) is of degree r then there is no g ∈ N

r−1,†k (N) having the same

p-adic q-expansions.

Proof. We prove this by contradiction. Let us assume that such a g exists. Let h = f −g. Since f (q,X) is ofdegree r and g ∈Nr−1,†

k (N), h(q,X) is still of degree r and therefore h is non-zero. However, by assumptionh(q,0) = 0. This implies that h = 0 by the diagram of the previous proposition, which is a contradiction.ut


3.2.3 E2, Θ and overconvergence

In the following two corollaries, we recover the main results of [7] using the polynomial q-expansion prin-ciple. It can be easily generalized to modular forms for other Shimura varieties.

Corollary 2. The p-adic modular form E2 is not overconvergent.

Proof. By the Corollary 1, this is immediate since the polynomial q expansion of E2 is of degree 1. ut

Corollary 3. If f is overconvergent of weight k and k 6= 0, then Θ . f is not overconvergent.

Proof. It follows from Proposition 3 and Proposition 6, that Θ . f is the image of the nearly overconvergentform δk. f in the space of p-adic modular forms by the map (2). Moreover

(δk. f )(q,X) = qd

dqf + kX f (q)

It is therefore of degree 1 since k 6= 0 and the result follows from Corollary 1. ut

3.2.4 The overconvergent projection

We give a p-adic version of the holomorphic projector.

Lemma 2. Let f be a nearly overconvergent form of weight k and order ≤ r such that k > 2r. Then foreach i = 0, · · · ,r, there exists a unique overconvergent form gi of weight k−2i such that

f =r

∑i=1

δik−2i.gi

Proof. This is special case of the proof of Proposition 9 below. utWe define the overconvergent projectionH†( f ) of f by:

H†( f ) := g0

It is an overconvergent version of the holomorphic projection since if f is holomorphic, then we clearlyhave:

H†( f ) =H( f )

which means that H†( f ) is holomorphic.

Remark 7. Let f ∈ Nr,†k (N,Qp) and g ∈ N

s,†l (N,Qp) such that k+ l > 2s+ 2r. Then the following holds

H†( f δ ml g) = (−1)mH†(gδ m

k f ). One can also show that when a Hecke equivariant p-adic Petersson innerproduct is defined then δ and ε are very close to be adjoint operators. This implies a formula of the type〈 f ,g〉p−adic = 〈 f ,H†(g)〉p−adic when f is overconvergent. We hope to come back to this in a future paper.

3.2.5 Action of the Atkin-Hecke operator Up

If ρ > p−1/p+1, it follows from the theory of the canonical subgroup (Katz-Lubin) that we can extendcanonically ϕ on Xord into

ϕ : X≥ρ → X≥ρ p.

Let E/X≥ρ pbe the generalized universal elliptic curve over X≥ρ p

and let E(ϕ)/X≥ρ be its pullback by ϕ .We have degree p isogeny

EFϕ→ E(ϕ)

over X≥ρ and we denote Vϕ : E(ϕ)→ E the dual isogeny. On the level of sheaves, the operator Up is definedas the composition of the following maps.

Hrk/X≥ρ

V ∗ϕ−→Hr,(ϕ)k /X≥ρ =Hr

k/X≥ρ p ⊗ϕ∗ O/X≥ρ

Id⊗ 1p ·Tr→ Hr

k/X≥ρ pj→Hr

k/X≥ρ

296 Eric Urban

where j is induced by the completely continuous inclusion OX≥ρ p → OX≥ρ defined by the restriction ofanalytic function on X≥ρ p

to X≥ρ and Tr is induced by the trace of the degree p map ϕ∗ : OX≥ρ p → OX≥ρ .Since j is completely continuous, Up induces a completely continuous endomorphism of Nr,ρ

k (N,Qp). Thefollowing proposition is easy to prove.

Proposition 7. Let f ∈Nr,†k (N,Qp). Let us write its polynomial q-expansion as:

f (q,X) =∞

∑n=0

a(n, f )(X)qn

Then we have:

(i) ( f |Up)(q,X) = ∑∞n=0 a(np, f )(pX)qn

(ii)ε( f |Up) = p.(ε f )|Up(iii)(δk f )|Up = pδk( f |Up)

Proof. (i) follows from a standard computation and (ii) and (iii) follow from (i) and the effect of δk and ε

on the polynomial q-expansion explained in Section 2. ut

Let N∞,ρk (N,Qp) be the p-adic completion of

N∞,ρk (N,Qp) :=

⋃r≥0

Nr,ρk (N,Qp).

Then we have:

Corollary 4. The action of Up on N∞,ρk (N,Qp) is completely continuous.

Proof. It follows easily from the lemma below for the sequence Mi = Ni,ρk (N) and the relation (ii) of the

previous proposition. ut

Lemma 3. Let Mi be an increasing sequence of Banach modules over a p-adic Banach algebra A. Let u bean endomorphism on M :=

⋃i Mi such that

(i) u induces a completely continuous endomorphism on each of the Mi’s.(ii)Let αi be the norm of the operator on the Hausdorff quotient of Mi/Mi−1 induced by u. Then the sequence

αi converges to 0.

Then u induces a completely continuous operator on the p-adic completion of M.

Proof. This is an easy exercise which is left to the reader. ut

Remark 8. We can give a sheaf theoretic definition of N∞,ρk . Let A be the ring of analytic functions defined

over Qp on the closed unit disc. It is isomorphic to the power series in X with Qp-coefficient convergingto 0. We denote it Ak if one equips it with the representation of the standard Iwahori subgroup of SL2(Zp)defined by:

((

a bc d

). f )(X) = (a+ cX)k f (

b+dXa+ cX

)

If we restrict this representation to the Borel subgroup we get a representation that contains Zp[X ]r(k) forall r. In fact, it is the p-adic completion of Zp[X ](k) =

⋃r Zp[X ]r(k). It is not difficult to see, one can define

a sheaf of Banach spaces on X≥ρ (in the sense of [1]) by considering the contracted product

H∞k := T×B Ak

Then, we easily see thatN

∞,ρk = H0(X≥ρ ,H∞

k )


3.2.6 Slopes of nearly overconvergent forms

Let f ∈ Nr,†k (N,Qp) which is an eigenform for Up for the eigenvalue α . If λ 6= 0, we say f is of finite

slope α = vp(λ ). The following proposition compares the slope and the degree of near overconvergenceand extends the classicity result of Coleman to nearly overconvergent forms.

Proposition 8. Let f ∈N∞,†k (N,Qp), then the following properties hold

(i) If f is of slope α , then its degree of overconvergence r satisfies r ≤ α ,(ii)If f is of degree r and slope α < k−1− r, then f is a classical nearly holomorphic form.

Proof. The part (i) is easy. If r is the degree of near overconvergence of f , then g = εr. f is a non trivialoverconvergent form and by Proposition 7 its slope is α−r. Since a slope has to be non-negative, (i) follows.The part (ii) is a straightforward generalization of the result for r = 0 which is a theorem of Coleman [5]. Wemay assume that r is the exact degree of near overconvergence. By the point (i), we therefore have α ≥ r.From the assumption, we deduce k−1− r ≥ r. Therefore k > 2r and we may apply Lemma 2:

f =r

∑i=o

δik−2igi

with gi overconvergent of weight k− 2i for each i. By uniqueness of the gi’s, we see easily that that theδk−2igi are eigenforms for Up with the same eigenvalue as f . So for each i, gi is of slope α − i < k− 1−r− i ≤ (k−2i)−1. Therefore it is classical by the theorem of Coleman. This implies f is classical nearlyholomorphic. ut

3.3 Families of nearly overconvergent forms

3.3.1 Weight space

Let X be the rigid analytic space over Qp such that for any p-adic field L ⊂ Qp X(L) = Homcont(Z×p ,L×).Any integer k ∈ Z can be seen as the point [k] ∈X(Qp) defined as [k](x) = xk for all x ∈ Z×p . Recall we havethe decomposition Zp = ∆ ×1+qZp where ∆ ⊂ Z×p is the subgroup of roots of unity contained in Z×p andq = p if p odd and q = 4 if p = 2. We can decompose X as a disjoint union

X=⊔

ψ∈∆

Bψ

where ∆ is the set of characters of ∆ and Bψ is identified to the open unit disc of center 1 in Qp. If κ ∈X(L)then it correspond to uκ ∈ Bψ(L) if κ|∆ = ψ and κ(1+q) = uκ .

3.3.2 Families

Let U ⊂ X be an affinoid subdomain. It is known from the works of Coleman-Mazur [6] and Pilloni [18],that there exist ρU ∈ [0,1)∩ pQ such that for all ρ ≥ ρU, there exists an orthonomalizable Banach spaceM

ρ

U over A(U) such that for all κ ∈ U(Qp), we have

Mρ

U⊗κ Qp∼= H0(X≥ρ ,ωκ)

We consider the sheaf7 ΩU over U×X≥ρ associated to the A(U×X≥ρ)-module Mρ

U and we put

HrU := ΩU⊗Hr

0

For any weight k ∈ Z such that [k] ∈ U(Qp), we recover Hrk by the pull-back

7 In [18], this sheaf is constructed in a purely geometric way and the existence of Mρ

U is deduced from it.

298 Eric Urban

([k]× idX≥ρ)∗Hr

U∼=Hr

k (1)

For general weight κ ∈ U(L), we define the sheaf Hrκ = ωκ ⊗Hr

0 where ωκ is the invertible sheaf definedin [18, §3]. We define the space of nearly overconvergent forms of weight κ by

Nr,ρκ (N,L) := H0(X≥ρ ,Hr

κ /L)

and the space of U-families of nearly overconvergent forms:

Nr,ρU (N) := H0(X≥ρ ,Hr

U)

We also define Nr,†κ (N) and N

r,†U (N) the spaces we obtain by taking the inductive limit over ρ . The space

Nr,ρU (N) is a Banach module over A(U) and for any weight κ ∈ U(L), we have

Nr,ρU (N)⊗κ L =N

r,ρκ (N,L)

This follows easily from (1) and the fact that X≥ρ is an affinoid.As in the previous section, one can define an action of Up on these spaces and show it is completely

continuous. For any integer r, any affinoid U⊂ X and ρ ≥ ρU, we may consider the Fredholm determinant

PrU(X) = Pr

U(κ,X) := det(1−X .Up|Nr,ρU ) ∈ A(U)[[X ]]

because one can show as in [18] that Nr,ρU (N) is A(U)-projective. A standard argument shows this Fredholm

determinant is independent of ρ . For U= κ, we just write Prκ(X). If r = 0, we omit r from the notation.

For any integer m, we consider the map [m] : X→ X defined by κ 7→ κ.[m] and we denote U[m], theimage of U by this map. We easily see from its algebraic definition, that the operator ε can be defined infamilies and induces a short exact sequence:

0→Mρ

U→Nr,ρU →N

r−1,ρU[−2]→ 0

From Proposition 7 and the above exact sequence one easily sees by induction on r that

PrU(κ,X) =

r

∏i=0

PU[−2i](κ.[−2i], pi.X)

Let us define N∞,ρU (N) as the p-adic completion of N∞,ρ

U (N) :=⋃r≥0

Nr,ρU (N). Then it follows from Lemma

3 again that Up acts completely continuously on it with the Fredholm determinant given by the convergingproduct

P∞U (κ,X) =

∞

∏i=0

PU[−2i](κ.[2i], piX)

Definition 1. Let U⊂ X be an affinoid subdomain and Q(X) ∈ A(U)[X ] be a polynomial of degree d suchthat Q(0) = 1. The pair (Q,U) is said admissible for nearly overconvergent forms (reps. for overconvergentforms) if there is a factorization

P∞U (X) = Q(X)R(X) (resp. PU(X) = Q(X)R(X) )

with P and Q relatively prime and Q∗(0) ∈ A(U)× with Q∗(X) := XdQ(1/X).

If (Q,U) is admissible for nearly overconvergent forms, it results from Coleman-Riesz-Serre theory [4]that there is a unique Up-stable decomposition

N∞,ρU =NQ,U⊕SQ,U

such that NQ,U is projective of finite rank over A(U) with

(i) det(1−UpX |NQ,U) = P(X)(ii)Q∗(Up) is invertible on SQ,U


It is worth noticing also that the projector eQ,U of N∞,ρU onto NQ,U can be expressed as S(Up) for some entire

power series S(X)∈XA(U)X . If we have two admissible pairs (Q,U) and (Q′,U′), we write (Q,U)< (Q′,U′)if U ⊂ U′ and if Q divides the image Q′|U(X) of Q′(X) by the canonical map A(U′)[X ]→ A(U)[X ]. Whenthis happens, we easily see from the properties of the Riesz decomposition that

eQ,U (eQ′,U′ ⊗A(U′) 1A(U)) = eQ,U (2)

We have dropped ρ from the notation in NQ,U since this space is clearly independant of ρ by a standardargument. We define N

f sU as the inductive limit of the NQ,U over the Q’s. Since NQ,U is of finite rank and

Up-stable it is easy to see that there exists r such that

NQ,U ⊂Nr,†U

Remark 9. If αQ,U is the maximal8 valuation taken by the values of the analytic function Q∗(0) ∈ A(U) onU, then one can easily see that r ≤ αQ,U by the point (i) of Proposition 8.

3.3.3 Families of q-expansions and polynomial q-expansions

By evaluating at the Tate object we have defined in section 2, w e can define the polynomial q-expansion ofan element F ∈ N

r,ρU (N) that we write F(q,X) ∈ A(U)[X ]r[[q]]. The evaluation Fκ(q,X) at κ of F(q,X) is

the polynomial q-expansion of the nearly overconvergent form of weight κ obtained by specializing F at κ .We also denote Fκ(q) = Fκ(q,0) the p-adic q-expansion of the specialization of F at κ . In what follow, weshow that when the slope is bounded a family of q-expansion of nearly overconvergent forms is equivalentto a family of nearly overconvergent forms.

Let F(q) ∈ A(U)[[q]] and Σ ⊂ U(Qp) a Zariski dense subset of points. We say that F(q) is a Σ -family ofq-expansions of nearly overconvergent form of type (Q,U) if for all but finitely many κ ∈ Σ the evaluationFκ(q) of F(q) at κ is the p-adic q-expansion of a nearly overconvergent form of weight κ and type Qκ

(i.e. is annihilated by Q∗κ(Up)). Let NΣQ,U be the A(U)-module of families of q-expansion of nearly overcon-

vergent forms of type (Q,U). Similarly, we can define NΣ ,polQ,U ⊂ A(U)[X ][[q]] the subspace of polynomial

q-expansion satisfying a smiler property for specialization at points in Σ with an obvious map:

NΣ ,polQ,U →NΣ

Q,U

given by the evaluation X at 0. Then we have:

Lemma 4. The q-expansion map and polynomial expansion maps induce the isomorphisms

NQ,U∼=N

Σ ,polQ,U

∼=NΣQ,U.

Proof. From Proposition 6, it suffices to show that the q-expansion map induces:

NQ,U∼=NΣ

Q,U.

The argument to prove this is well-known but we don’t know a reference for it. We therefore sketch it below.Notice first that for any κ0 ∈ U(Qp) the evaluation map at κ0 induces an injective map:

NΣQ,U⊗κ0 Qp →Qp[[q]] (3)

Indeed if F ∈ NΣQ,U is such that Fκ0(q) = 0 then if ϖκ0 ∈ A(U) is a generator of the ideal of the elements

of A(U) vanishing at κ0, we have F(q) = ϖκ0 .G(g) for some G ∈ A(U)[[q]]. Clearly for any κ ∈ Σ\κ0,we have Gκ(q) = 1

ϖκ0 (κ)Fκ(q) is the q-expansion of a nearly over convergent for of weight κ and type Qκ .

Therefore G ∈ NΣQ,U and our first claim is proved. Now let κ ∈ Σ . We have the following commutative

diagram:

8 This maximum is < ∞ since Q∗(0) ∈ A(U)×

300 Eric Urban

NQ,U⊗κ Qp(3) //

(1)

NΣQ,U⊗κ Qp

(2)

NΣQκ ,κ

(4) // Qp[[q]]

Since (2) and (4) are injectives and (1) is an isomorphism, we deduce (3) is injective. Now since the imageof (2) is included in the image of (4) and (1) is surjective, we deduce that (3) is an isomorphism of finitevector spaces. Since NΣ

Q,U is torsion free over A(U) such that NΣQ,U⊗κ Qp has bounded dimension when

κ runs in Σ , a standard argument shows that NΣQ,U is of finite type over A(U) (see for instance [26, §1.2]).

Notice that the injectivity of (3) below is true for all κ and therefore we deduce that the map

NQ,U→NΣQ,U (4)

is injective with a torsion cokernel of finite type. We want now to prove the surjectivity. Let F(q) ∈ NΣQ,U

and let IF ⊂ A(U) be the ideal of element a such that a.F(q) is in the image of (4) and let aF be a generatorof IF . Let G ∈NQ,U whose image is aF .F(q). For any κ0 such that aF(κ0) = 0, we get that Gκ0 = 0. By theisomorphism (1), G is therefore divisible by ϖκ0 and thus aF

ϖκ0∈ IF which contradicts the fact that aF is a

generator of IF . Therefore aF does not vanish on U and F is in the image of (4). This proves the surjectivitywe have claimed. ut

More generally, for any Qp-Banach space M, we can define NΣQ,U(M) the subspace of elements F ∈

A(U)⊗M[[q]] such that for almost all κ ∈ Σ , the evaluation Fκ at κ of F(q) is the q-expansion of an elementof NQκ ,κ(M) =NQκ ,κ ⊗M. Similarly, one defines NΣ ,pol

Q,U (M). Then it is easy to deduce the following:

Corollary 5. We have the isomorphisms:

NQ,U⊗M ∼=NΣ ,polQ,U (M)∼=NΣ

Q,U(M).

Proof. Left to the reader. ut

3.3.4 Maass-Shimura operator and overconvergent projection in p-adic families

The formula for the action of the Maass-Shimura operators on the q-expansion suggests it behaves well infamilies. We explain this here using the lemma 4. This could be avoided but it would take more time thanwe want to devote to this here. We explain this in a remark below.

We defined the analytic function Log(κ) on X by the formula:

Log(κ) =logp(κ(1+q)t)

logp((1+q)t)

where logp is the p-adic logarithm defined by the usual Taylor expansion logp(x) = −∑∞n=1

(1−x)n

n for allx∈Cp such thar |x−1|p≤ p−1 and t is an integer greater than 1/vp(κ(1+q)). Of course, from the definitionwe have Log([k]) = k. Moreover Log is clearly an analytic function on X.

If F(q,X) = ∑∞n=0 an(X ,F)qn ∈ A(U)[X ]r[[q]], we define

δF(q,X) := D.F(q,X)+Log(κ)XF(q,X)

where

D = q∂

∂q−X2 ∂

∂X.

If F(q,X) ∈ NΣ ,polQ,U for a Zariski-dense set of classical weight Σ , it is clear that δ .F(q,X) ∈ N

Σ ,polQ,U

with

Σ = Σ [2] and Q(κ,X) = Q(κ.[−2], pX). We therefore thanks to Lemma 4 deduce we have a map

δ : N†,r,fsU →N

†,r+1,fsU[2]


Remark also, it is straightforward to see that the effect of the operator ε on the polynomial q-expansion offamilies is the partial differentiation with respect to X :

(ε.F)(q,X) =∂

∂XF(q,X) ∀F ∈N

†,r,fsU

Remark 10. Like in Remark 8, we can give a sheaf theoretic definition of N∞,ρU . For simplicity, let us assume

that all the p-adic characters in U are analytic on Zp. Let AU := A(U)⊗A. Elements in AU, can be seen asrigid analytic functions on U×Zp. It is equipped with the representation of the standard Iwahori subgroupof SL2(Zp) defined by:

((

a bc d

). f )(κ,X) = κ(a+ cX) f (

b+dXa+ cX

)

Again as in remark 8, one can define but now using the technics of [18] a sheaf of Banach spaces on X≥ρ×U

H∞U := T×B AU

∼= ωU⊗H∞0

and show that we have:N

∞,ρU = H0(X≥ρ ×U,H∞

U)

Since AU is a representation of the Lie algebra of sl2, it would be possible to define a connection using theBGG formalism like in the algebraic case (see for instance [25, §3.2])

H∞U→H∞

U[2]

One would then obtain the Maass-Shimura operator in family without the finite slope condition:

δ : Nρ,∞U → N

ρ,∞U[2]

It is then easy to verify that δ (Nρ,rU )⊂N

ρ,r+1U[2] . We leave the details of this construction for another occasion

or to the interested reader.Finally, we want to mention that Robert Harron and Liang Xiao [27] have also given a geometric con-

struction of this operator in family using a splitting of the Hodge filtration and showing the definition isindependent of the chosen splitting. The above sketched construction can be done without such a choice butit probably boils down to a similar argument.

Now we have the following proposition.

Proposition 9. Let U⊂X be an open affinoid subdomain and F ∈N†,r,fsU , then for each i ∈ 0, . . . ,r there

existsGi ∈

1

∏2rj=2(Log(κ)− j)

N†,0,fsU[−2i]

such thatF = G0 +δ .G1 + · · ·+δ

r.Gr

Moreover, this decomposition is unique.If U= κ such that Log(κ) 6∈ 2,3, . . .2r, the result holds as well.

Proof. It is sufficient to prove this when U is open since we can obtain the general result after specialization.We prove this by induction on r. Notice that for G ∈N

†,0,fsU[−2r], we have by (1)

εr.δ r.G = r!.

r

∏i=1

(Log(κ)− r− i).G

since the left hand and right hand sides coincide after evaluation at classical weights bigger than 2r. We put

Gr :=1

r!.∏ri=1(Log(κ)− r− i)

εrF

302 Eric Urban

then Gr ∈ 1∏

ri=1(Log(κ)−r−i)N

†,0,fsU[−2r] and F−δ r.Gr is by construction of degree of nearly overconvergent less

or equal to r−1. We conclude by induction. utThen we define

H†(F) := G0

This is the overconvergent (or holomorphic) projection in family since it clearly coincides with the holo-morphic projection for nearly holomorphic forms of weight k > 2r.

Lemma 5. For any nearly overconvergent family of finite slope F ∈N†,r,fsU , and Heke operator T , we have

H†(T.F) = T.H†(F).

In particular, for any admissible pair (Q,U) and we have

eQ,U(H†(F)) =H†(eQ,U(F)).

Proof. It follows easily from the relation δ j(T (n).F) = n jT (n).δ j(F) and the uniqueness of the Gi’s in thedecomposition of Proposition 1. ut

4 Application to Rankin-Selberg p-adic L-functions

Let E be the eigencurve of level 1 constructed by Coleman and Mazur. in [6]. In this section, we givethe main lines of a construction of a p-adic L-function on E×E×X. The general case of arbitrary tamelevel can be done exactly the same way. The restriction of our p-adic L-function to the ordinary part ofthe eigencurve, gives Hida’s p-adic L-function constructed in [13] and [14]. Our method follows closelyHida’s construction for ordinary families of eigenforms. We are able to treat the general case using theframework of nearly overconvergent forms. We will omit the details of computation that are similar toHida’s construction and will focus on how we get rid of the ordinary assumptions. We don’t pretend to anyoriginality here. We just want to give an illustration of the theory of nearly overconvergent forms to theconstruction of p-adic L-functions in the non-ordinary case.

4.1 Review on Rankin-Selberg L-function for elliptic modular forms

We recall the definition and integral representations of the Rankin-Selberg L-function of two elliptic mod-ular forms and its critical values. Let f and g be two elliptic normalized newforms of weights k and l withk > l and nebentypus ψ and ξ respectively of level M. We denote their Fourier expansion by:

f (z) =∞

∑n=1

anqn and g(z) =∞

∑n=1

bnqn

Shimura is probably the first one to study in [20] the algebraicities of the critical values of

DM(s, f ,g) := L(ψξ ,k+ l−2s−2)(∞

∑n=1

anbnn−s)

More precisely he proved that for every integer m ∈ 0, . . . ,k− l−1, then

D(l +m, f ,g)π l+2m+1〈 f , f 〉M

∈Q

Here 〈 f , f 〉M is the Petersson inner product of f with itself. Recall it is defined by the formula

〈 f ,g〉M =∫

Γ1(M)\hf (τ)g(τ)yk−2dxdy


When 0≤ 2m < k− l, the essential ingredient in the proof of Shimura was to establish a formula of thetype

DM(l +m, f ,g) = 〈 f ,gδmk−l−2mE〉M = 〈 f ,H(gδ

mk−l−2mE)〉M

where E is a suitable holomorphic Eisenstein series of weight k− l−2m.When f and g vary in Hida families and m is also allowed to vary p-adically, Hida has constructed

a 3-variable p-adic L-function interpolating a suitable p-normalization these numbers. We now recall theprecise formula that is used to interpolate these special values in [13]. We first need some standard notations.For any integer M, we put τM =

(0M−10

)and for any modular form h, we dente by hρ the form defined

by hρ(τ) = h(−τ) for τ ∈ h. For any Dirichlet character χ of level M and any integer j ≥ 2 such thatχ(−1) = (−1) j, we denote by E j(χ) the Eisenstein series of level M, nebentypus χ and weight j whoseq-expansion is given by

E j(χ)(τ) =L(1−m,χ)

2+

∞

∑n=1

( ∑d|n

(d,M)=1

χ(d)d j−1)qn

Proposition 10. [13, Thm 6.6 ] Let L be an integer such that f and g are of level Lpβ , then we have

DLp(l +m, f ,g) = t.π l+2m+1〈 f ρ |kτLpβ ,H(g|lτLpβ δmk−l−2m(Ek−l−2m,Lp(ψξ )〉Lpβ

with

t =2k+l+2m(Lpβ )

k−l2 −m−1il−k

m!(l +m−1)!.

and m ∈ Z with 0≤ m < (k− l)/2.

4.2 The p-adic Petersson inner product

For simplicity, we assume the tame level is 1. Fix and admissible pair (R,V) for overconvergent forms andlet MR,V the corresponding associated space of V-families of overconvergent forms. Let TR,V the Heckealgebra acting on MR,V. A standard argument using the q-expansion principle shows that the pairing

MR,V⊗A(U) TR,V→ A(V)

given by(T, f ) := a(1, f |T )

is a perfect duality. Since the level9 is 1, we also know that TR,V is reduced. Therefore, the trace mapinduces a non-degenerate pairing on TQ,U with ideal discriminant dR,V ⊂ A(V) whose set of zeros is theset of weight where the map EQ,U→ U is ramified. In particular, we have a canonical isomorphism:

MR,V⊗F(V)∼= TR,V⊗F(V) (1)

4.2.1

From this, we deduce a Hecke-equivariant pairing

(−,−)R,V : MR,V⊗A(V)⊗MR,V −→ F(V)

Let now F be a Galois extension of F(V) the field of fraction of A(V). We assume that for each irreduciblecomponent C of ER,V, F contains the function field F(C) of C. For each irreducible component C, we definethe corresponding idempotent 1C ∈ TR,V⊗F and we write FC for the element defined by

9 When the level is not 1, one uses the theory of primitive forms which described the maximal semi-simple direct factor ofTR,V⊗F(V)

304 Eric Urban

FC :=∞

∑n=1

λC(Tn)qn ∈ F[[q]]

where λC is the character of the Hecke algebra defined by T.1C ⊗ idF = 1C ⊗ λC(T ). If we denote by(−,−)F the scalar extension of (−,−)R,V to F then the Hecke invariance of the inner product implies that

a(1,1C.G) = (FC,G)F

is the coefficient of FC when one writes G as a linear decomposition of the eigen families FC’s.

Remark 11. This construction can be easily extended to the space NR,V for any admissible psi (R,V) fornearly overconvergent forms.

4.3 The nearly overconvergent Eisenstein family

We consider the Eisenstein family E(q) ∈ A0(X)[[q]] such that for each weight κ ∈X(Qp), its evaluation atκ is given by

E(κ,q) =∞

∑n=1

a(n,E,κ)qn :=∞

∑n=1

(n,p)=1

∑d|n〈d〉κ .d−1qn.

where for any m ∈ Z×p , 〈m〉κ ∈ A(X) stands for the analytic function of X defined by

κ 7→ κ(m)

In particular, when κ = [k].ψ with ψ a ramified finite order character of Z×p , then E(κ,q) is the q-expansionof

E(p)k (ψ)(τ) := Ek(ψ)(τ)−Ek(ψ)(pτ).

We define the nearly overconvergent Eisenstein family Θ .E ∈ A0(X×X)[[q]] by

Θ .E(κ,κ ′) :=∞

∑n=1

(n,p)=1

〈n〉κ a(n,E,κ ′)qn

Lemma 6. If κ = [r] and κ ′ = [k]ψ , the evaluation at (κ,κ ′) of Θ .E is

Θ .E(κ,κ ′) =Θr.E(p)

k (ψ)(q).

It is the p-adic q-expansion of the nearly holomorphic Eisenstein series δ rk E(p)

k (ψ).

Proof. The first part is obvious and the second part follows from the formula (4) and the canonical diagramof Proposition 6. ut

4.4 Construction of the Rankin-Selberg L-function on E×E×X

4.4.1 Some preparation

Let (Q,U) be an admissible pair for overconvergent forms of tame level 1 and let TQ,U be the correspondingHecke algebra over A(U). By definition it is the ring of analytic function on the affinoid subdomain EQ,U

sitting over the affinoid subdomain ZQ,U associated to (Q,U) of the spectral curve of the Up-operator. Recallthat

ZQ,U = Max(A(U)[X ]/Q∗(X))⊂ ZUp ⊂ A1rig×U


where ZUp is the spectral curve attache to Up (i.e. the set of points (α,κ) ∈ A1rig×U such that P0

κ (α) = 0)and

TQ,U = A(EQ,U) with EQ,U = E⊗ZUpZQ,U

The universal family of overconvergent modular eigenforms of type (Q,U) is given by

GQ,U :=∞

∑n=1

T (n)qn ∈ TQ,U[[q]]

Tautologically, for any point x ∈ EQ,U of weight κx ∈ U, the evaluation GQ,U(x) at x of GQ,U is the overcon-vergent normalized eigenform fx of weight κx associated to x.

LetGE

Q,U := GQ,U.Θ .E ∈ TQ,U⊗Ab(X×X)[[q]] = Ab(EQ,U×X×X)[[q]]

The Fourier coefficients of this Fourier expansions are analytic functions on EQ,U×X×X. Let now (R,V)be an admissible pair for nearly overconvergent forms of tame level 1. Then we consider

GEQ,U,R,V(q) ∈ Ab(V×EQ,U×X)[[q]]

defined byGE

Q,U,R,V(κ,y,ν)(q) := eR,V.GEQ,U(y,ν ,κκ

−1y ν

−2)(q)

and where eR,V = S(Up) for some S ∈ X .A(V)[[X ]] is the projector of N∞,fsV onto NR,V.

Proposition 11. With the notation above GEQ,U,R,V(q) is the q-expansion of an element of GE

Q,U,R,V ∈NR,V⊗Qp Ab(EQ,U×X). Moreover if we have (Q,U) < (Q′,U′) and (R,V) < (R′V′), then GE

Q,U,R,V isthe image of GE

Q′,U′,R′,V′ by the natural map

NR′,V′ ⊗Qp Ab(EQ′,U′ ×X)→NR,V⊗Qp Ab(EQ,U×X)

Proof. By Corollary 5, with with (Q,U) replaced by (R,V) and with M = Ab(EQ,U×X), it is sufficient toshow that the specialization at a Zariski dense set of arithmetic points of ([k],x, [r]) ∈V×EQ,U×X is theq-expansion of a nearly holomorphic form of weight k annihilated by R∗[k](Up). It is sufficient to choose thetriplet ([k],x, [r]) such that κx = [l] with l ∈ Z≥2, r ≥ 0 such that k− l− 2r ≥ 0 since such triplets form aZariski dense set of V×EQ,U×X. The evaluation at such a triplet is easily seen to be the p-adic q-expansionof eR,k(gx.Θ

r.Ek−l−2r). By definition of eR,k it follows that this form belongs to NR,k. The second part of theproposition is a trivial consequence of (2). ut

4.4.2 A 3-variable p-adic meromorphic function

Let ZR,V ⊂ A1rig ×V the affinoid of the spectral curve Z∞

Upattached to P∞

V. This affinoid is a priori notcontained in the spectral curve attached to Up but the eigencurve is still sitting over it since ZUp ⊂ Z∞

Up. We

can therefore considerER,V = EN×Z∞

U−pZR,V

and the ER,V’s form an admissible covering of EN when the (R,V) vary.Let C⊂ ER,V be an irreducible component. Then we set

DC,Q,U := a(1,1C.H†(GEQ,U,R,V)) ∈ F(C)⊗F(EQ,U×X)

Remark 12. If HC ⊂ A(C) is a denominator of 1C, then the poles of DC,Q,U come from the zeros of HC andthe poles of the overconvergent projector. Therefore we have:

HC.

2rR,V

∏i=2

(Log(κ)− i).DC,Q,U ∈ A(C×EQ,U×X) (1)

We denote by DR,V,Q,U the unique element of

306 Eric Urban

F(ER,V×EQ,U×X) = ∏C⊂ER,Virreducible

F(C)⊗F(EQ,U×X)

restricting to DC,Q,U on C×EQ,U×X for each irreducible component C of ER,V. It can be constructed asthe image of H†(GE

Q,U,R,V) in TR,V⊗F(V)⊗F(EQ,U×X) = F(ER,V×EQ,U×X) by the map (1) tensoredby F(EQ,U×X)

We have the following result:

Lemma 7. There exist a meromorphic function D on E×E×X whose restriction to F(ER,V×EQ,U×X)gives DR,V,Q,U for any quadruplet (R,V,Q,U).

Proof. If we have pairs (R,V),(Q,U),(R′,V′),(Q′,U) with (R,V) < (R′,V′) and (Q,U) < (Q′,U′) thenwe have by (2) eR′,V′ |V eR,V = eR,V and eQ′,U′ |U eQ,U = eQ,U. Since the overconvergent projection isHecke equivariant, we deduce that

eR,V⊗ eQ,V.(DR′,V′,Q′,U′ |V×U) = DR,V,Q,U

and that the DR,V,Q,U’s glue to define a memormorphic function D ∈ F(E×E×X).

4.4.3 The interpolation property

For x ∈ E(Qp), we denote θx the corresponding character of the Hecke algebra. If x is attached to a classicalform, we denote by fx the eigenform attached to x. By definition,

ιp ι−1∞ ( fx(q)) =

∞

∑n=1

θx(Tn)qn

We denote by kx its weight, ψx its nebentypus and pmx its minimal level with mx a positive integer. We willalways assume kx ≥ 2 and that pmx is the conductor of ψx. We consider the complex number W ( fx) definedby

hx := f ρx |τpmx =W ( fx) fx

It is a complex number of norm 1 called the root number of fx.If E is smooth at x, then there is only one irreducible component containing it and if C is the irreducible

component of an affinoid ER,V containing x, then

HC(κx) 6= 0 (2)

In that case, we can define the specialization 1x ∈ TRκx ,κx of 1C at κx and it satisfies:

Tn.1x = θx(Tn).1x ∀n

In general, TRκx ,κx is not semi-simple so 1x is not necessarily the (generalized) θx-eigenspace projector. Butif the projection map E→ X is étale at x then it is. We know it is the case when x is non-critical; recall thatx is said non-critical if

vp(θx(Up))< kx−1.

If all the slopes of Rκx are strictly less that kx−1 (something that we can assume after shrinking ER,V), theimage of 1C into the Hecke algebra acting on the space of forms MRκx ,κx is the projector 1 fx attached to thenew form fx. Moreover we can show by the same computations as [12, sect. 4] that

a(1,1 fx .g) = a(p, fx)mx−n.p(n−mx)(kx/2−1) 〈 f

ρx |τpn ,g〉pn

〈hx, fx〉pmx(3)

for any g ∈MRκx ,κx of level pn with n≥ mx.For any ν ∈ X(Qp), we write kν := log(ν). We say ν is arithmetic if kν ∈ Z and we denote ψν the finite

order character such that ν = [kν ].ψν .We have the following theorem.


Theorem 1. Let L be a finite extension of Qp and (x,y) ∈ E×E(L) and any arithmetic ν = [r].ψν such thatκx, κy are arithmetic and satisfy the following

(i) kx− ky > r ≥ 0,(ii)x is classical and non-critical,(iii)y is classical,(iv)The level of f ρ

y equals the level of f ρy |ψν

(v)ψx and ψy are ramified.

Then we have

D(x,y,ν) = (−1)kyW ( fx)W ( f ρy )a(p, fx)

mx−my pmx(1− kx2 )+my(r+

ky2 ) (4)

×Γ (ky + r)Γ (kx +1)Dpn( fx, f ρ

y |ψν ,ky + r)(2iπ)kx+ky+2r+1π1−kx〈 fx, fx〉pmx

Proof. This computation follows closely those of Hida in [14]. We treat the case kx−ky > 2r≥ 0. The casekx−ky > r≥ (kx−ky)/2 can be treated similarly (see for instance [14]) and is obtained using the functionalequation for the nearly holomorphic Eisenstein series. We also assume ψν is trivial to lighten the notations.

By our hypothesis, we can choose a quadruplet (R,V,Q,U) such that

(a)(x,y) ∈ ER,V×EQ,U(L)(b)The eigenvalues of R∗κx(X) ∈ L[X ] are of valuation smaller than kx−1

Then, D(x,y, [r]) = DR,V,Q,U(x,y, [r]). By the condition (b), we know that TRκx ,κx = TR,V⊗κx L is semi-simple and therefore the map ER,V → V is étale at x. In particular, ER,V is smooth at x and x belongs toonly one irreducible component C of ER,V. By construction, we have:

DR,V,Q,U(x,y, [r]) = a(1,1x H†(eRκx ,κx(GQ,U(y)ΘE([r],κxκ−1y [2r]−1))))

= a(1,1 fx(H†(eRκx ,κx( fyδ

rkx−ky−2rE

(p)kx−ky−2r))))

= a(1,1 fx eRκx ,κx(H†(g)))

with g = fyδ rkx−ky−2rE

(p)kx−ky−2r(ψxψ−1

y ). Since g is nearly holomorphic of order ≤ r and weight kx > 2r , we

have H†(g) =H(g) is holomorphic. Since H(g) is an holomorphic form of level pn with n = Max(mx,my),we have

DR,V,Q,U(x,y, [r]) = a(1,1 fx eRκx ,κxH(g)) (5)

= a(p, fx)mx−n.p(n−mx)(

kx2 −1).

〈 f ρx |τpn ,H(g)〉pn

〈hx, fx〉pmx

= a(p, fx)mx−n.p(n−mx)(

kx2 −1).

〈 f ρx |τpn ,g〉pn

〈hx, fx〉pmx

As in [14], we now transform fy a little:

fy = (−1)ky fy|τpn |τpn = (−1)ky .pky2 (n−my). fy|τpmy [pn−my ]|τpn

= (−1)kyW ( f ρy ).p

ky2 (n−my). f c

y |[pn−my ]|τpn

By replacing this in the expression above we get for (5):

308 Eric Urban

a(p, fx)mx−n.(−1)kyW ( f ρ

y ).p(n−mx)(

kx2 −1).p

ky2 (n−my).× (6)

〈 f ρx |τpn , f c

y |[pn−my ]|τpnE(p)kx−ky−2r(ψxψ−1

y )〉pn

〈hx, fx〉pmx=

a(p, fx)mx−n.(−1)kyW ( f ρ

y ).pn(ky+r)+mx(1− kx

2 )−myky2 ×

(kx + ky + r+1)!r!.Dpn( fx, f ρ

y |[pn−my ],ky + r)πky+2r+12kx+ky+2riky−kx〈hx, fx〉pmx

Now using the fact that for p dividing M, we have:

DM( f ,g[pm],s) = a(p, f )m p−msDM( f ,g,s))

we deduce that (5) is equal to

a(p, fx)mx−my .(−1)kyW ( f ρ

y ).pmx(1− kx

2 )+my(r+ky2 )×

(ky + r−1)!r!.Dpn( fx, f ρ

y |[pn−my ],ky + r)πky+2r+12kx+ky+2riky−kx〈hx, fx〉pmx

and the specialization formula stated in the theorem follows. ut

Remark 13. a) This result is still true if x is classical and critical if it is not θ -critical. The condition (iv)and (v) are not necessary and could be removed at the expanse to modify the formula by adding some Eulerfactors at p.b) From the construction, we see that this meromorphic function has possible poles along certain hyper-surfaces of E×E×X corresponding to intersections of the irreducible components of the first variable andalso along certain hypersurfaces created by the overconvergent projection. This happens when the overcon-vergent form fx is at the same time the specialization of a family of overconvergent forms and a family ofpositive order nearly overconvergent forms. It is easy to see that implies x is θ -critical. In the next section,we review the definition of a θ -critical point and compute the residue of D when the weight map at thispoint is étale.

4.4.4 Residue at an étale θ -critical point

Let x1 ∈ E(L) of classical weight k1 ≥ 2 and slope k1− 1. We say that x is θ -critical if there exist x0 ofweight k0 = 2− k1 such that fx1 = Θ k1−1 fx0 . Here we denote fx0 the ordinary form of weight k0 = 2− k1attached to x0. We then write x1 = θ(x0). We have the following result.

Theorem 2. Let x0 and x1 as above. Assume that κ : E→X is étale at x1 = θ(x0) or equivalently that E issmooth at x0. Then the order of the pole of D(x,y,ν) at x1 is at most one and

Res|x=x1(D(x,y,ν)) =∏

k1−2j=0 (Log(νκy)− j)(Log(ν)− j)

(k1−1)!D(x0,y,ν [1− k1]) (7)

for all (y,ν) ∈ E×X

Proof. The fact that E κ→X is étale at x1 is equivalent to E smooth at x0 is well-known and follows from R.Coleman’s work.

We choose (R0,V0) such that x0 ∈ ER0,V0(L). Consider the pair (R1,V1) with R1(κ,X) = R0(κ[2−2k1], pk1−1X) and V1 =V0[2k1−2].

For i = 0,1, let Ci be the (unique) irreducible component of ERi,Vi containing xi and let consider Fi = FCithe corresponding Coleman family. Let G = GE

R1,V1,Q,U for some admissible pair (Q,U).Let F be an extension of F(V) as in §4.2.1. Then by definition of D = DC1,Q,U and of the overconvergent

projection, we have

G = D.F1 +D′δ k1−1F0 +H (8)


with some D′ ∈ F⊗Ab(EQ,U×X) and H ∈ NR1,V1 ⊗Ab(EQ,U×X) such that (H,Fi)F = 0 for i = 0,1where (−,−)F is the p-adic Petersson inner product defined in §4.2.1. Notice that by our hypothesis,F1(x1) = fx1 = δ k1−1F0(x0). Since G is regular at [k1] and F1 and δ k1−1F0 are the only families of nearlyoverconvergent forms of finite slope specializing to fx1 , this implies that D+D′ is regular at x1. Thereforein particular the order of the pole of D at x1 is the same as the order of the pole of D′ at x1 and

Res|x=x1(D(x,y,ν)) =−Res|x=x1(D′(x,y,ν)) (9)

From (8), we have

εk1−1G(κx,y,ν) =

k1−2

∏j=0

(2−Log(κx)+ j)D′(x,y,ν).F0 + εk1−1H (10)

Since the eigencurve is smooth at x0, this implies that ∏k1−2j=0 (2−Log(κx)+ j)D′ is regular at x = x1 and

therefore the pole of D′ at x1 is at most simple. Moreover, we get

Res|x=x1(D′(x,y,ν)) =

(−1)k1

(k1−1)!.a(1,1C0 .ε

k1−1G([k1],y,ν)) (11)

Now we want to evaluate εk1−1G(κ,y,ν). For any classical triplet (κ,y,ν) = ([k],y, [r]) with k−ky > 2r≥ 0and ψ = ψyψx1 , we deduce from the evaluation of (2) at X = 0 that

εk1−1G(x,y,ν)(q) = εk1−1eR1,V1 fyΘ

r.Ek−ky−2r(ψ)(q)

= eR0,V0εk1−1 fyΘr.Ek−ky−2r(ψ)(q)

= eR0,V0 fyεk1−1Θ r.Ek−ky−2r(ψ)(q)

=Γ (k−ky−r)r!

Γ (k−ky−r−k1+1)(r−k1+1)! eR0,V0 fyΘr−k1+1Ek−ky−2r(ψ)(q)

We deduce that

εk1−1G(κ,y,ν) =

k1−1

∏j=1

(Log(κκ−1y ν

−1)− j)(Log(ν)− j+1)GER0,V0,Q,U(κ[2−2k1],y,ν [1− k1])

since the left and right hand sides of the above have the same evaluations on a Zariski dense set of point ofX×EQ,U×X. Evaluating at κ = [k1] gives:

(12)

εk1−1G([k1],y,ν) =

k1−2

∏j=0

( j−Log(νκy))(Log(ν)− j)GER0,V0,Q,U([2− k1],y,ν [1− k1])

Sincea(1,1C0GE

R0,V0,Q,U([2− k1],y,ν)) =D(x0,y,ν)

for (y,ν) ∈ EQ,U×X, the formula (7) follows from (9), (11) and (12). ut

Remark 14. This residue formula has a flavor similar to the work of Bellaiche [2] in which it is proved thatthe standard p-adic L-function attached to a θ -critical point is divisible by a similar product of p-adic Log’s.

Remark 15. It is also possible to define a two variable Rankin-Selberg p-adic L-function interpolating thecritical values Dp( fx, fy,kx−1) by replacing Θ .E(κ,κ ′) by Eord(κ) in our construction where for κ = [k].ψκ

and k ∈ Z≥2 we have

Eord(κ) =L(1− k,ψ)

2+

∞

∑n=1

( ∑d|n

(n,p)=1

κ(d)d−1)qn.

310 Eric Urban

Since E(κ) has a pole at κ = [0], this two-variable p-adic L-function would have a pole along the hyper-suface defined by κx = κy. It should be easy to compute the corresponding residue and obtain a formulasimilar to the one of Hida’s Theorem 3 [14, p. 228].

Acknowledgements The author would like to thank Giovanni Rosso and Chris Skinner for interesting conversations duringthe preparation of this work and Vincent Pilloni for pointing out an error in a previous version of this text. He would like alsoto thank Pierre Colmez who encouraged him to write this note. He is also grateful to the organizers of the conference Iwasawa2012 held in Heidelberg for their invitation and for giving the opportunity to publish this paper in the proceedings of thisconference. This work was also lectured during the Postech winter school in january 2013. The author would like to thank theorganizers of this workshop for their invitation. Finally the author would like to thank the Florence Gould Foundation for itssupport when he was a Member at the Institute for Advanced Studies and when some part of this work was conceived.

References

1. F. Andreatta, A. Iovita and V. Pilloni, Families of Siegel modular forms, to appear in Annals of Math. .2. J. Bellaiche, Critical p-adic L-functions, Invent. Math. 189 (2012), no. 1, 1–60.3. P. Berthelot and A. Ogus , Notes on crystalline cohomology, Princeton University Press ; University of Tokyo Press,

Tokyo, 1978. vi+243 pp4. R. Coleman, p-adic Banach spaces and families of modular forms. Invent. Math. 127 (1997), no. 3, pp 417–479.5. R. Coleman, Overconvergent and classical modular forms, Invent. math. 124 (1996), pp 215–241.6. R. Coleman and B. Mazur, The Eigencurve, Galois representations in arithmetic algebraic geometry (Durham, 1996),

vol. 254 of London Math. Soc. Lect. Note Ser. 1-113. Cambridge Univ. Press, Cambridge, 1998.7. R. Coleman, F. Gouvea and N. Jochnowitz, E2, Θ and overconvergence, IMRN International Mathematics Research

Notices 1995, No. 18. H. Darmon and V. Rotger, Diagonal cycles and Euler systems I: A p-adic Gross-Zagier Formula, to appear to the Ann.

Sc. Ecole Norm. Sup.9. M. Harris, Arithmetic vector bundles and automorphic forms on Shimura varieties. I, inventiones math. 82 (1985), pp

151-18910. M. Harris, Arithmetic vector bundles and automorphic forms on Shimura varieties. II, Compositio math., tome 60 (1986),

pp 323–378.11. Robin Hartshorne, Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard

1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20. Springer-Verlag, Berlin,12. H. Hida, A p-adic measure attached to the zeta functions associated with two elliptic modular forms. I, Invent. math. 79,

159–195 (1985)13. H. Hida, A p-adic measure attached to the zeta functions associated with two elliptic modular forms. II, Annales de

l’Institut Fourier, tome 38 n3, pp 1–83 (1988).14. H. Hida, Elementary theory of L-functions and Eisenstein series, Cambridge Univ. Press, London Math. Soc. Student

Texts 26, (1993).15. N. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes Ctudes

Sci. Publ. Math. No. 39 (1970), 175Ð232.16. N. Katz, p-adic Properties of Modular Schemes and Modular Forms, International Summer School on Modular Func-

tions, Antwerp (1972).17. M. Nappari, Holomorphic Forms Canonically Attached to Nearly Holomorphic Automorphic Forms Thesis Brandeis

University (1992).18. V. Pilloni, Formes modulaires surconvergentes , to appear in the Annales de l’Institut Fourier.19. G. Shimura, Arithmeticity in the Theory of Automorphic Forms, American Mathematical Society, Mathematical Surveys

and Monographs, Vol. 82, (2004).20. G. Shimura, The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (1976), no.

6, 783Ð-804.21. G. Shimura, IIntroduction to the arithmetic theory of automorphic functions, KanZ Memorial Lectures, No. 1. Publi-

cations of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press,Princeton, N.J., 1971. xiv+267 pp.

22. C. M. Skinner and E. Urban, p-adic families of nearly holomorphic forms and Applications, in preparation.23. E. Urban, Eigenvarieties for reductive groups, Annals of Math. 174 (2011), pp 1685–1784.24. E. Urban, On the ranks of Selmer groups of elliptic curves over Q, in Automorphic Representations and L-Functions,

Proceedings of the International Colloquium held at the Tata Institute of Fundamental Research, pp 651– 680 (2013)25. J. Tilouine, Companion forms and classicity in the GL2(Q)-case, Number theory, 119Ð141, Ramanujan Math. Soc. Lect.

Notes Ser., 15, Ramanujan Math. Soc., Mysore, 2011.26. A. Wiles, On ordinary λ -adic representations associated to modular forms, Invent. Math. 94 (1988), no. 3, 529Ð573.27. R. Harron and L. Xiao, Gauss-Manin connections for p-adic families of nearly overconvergent modular forms,

arXiv:1308.1732.

Noncommutative L-functions for varieties over finite fields

Malte Witte

Abstract We consider certain perfect complexes of adic sheaves on varieties over finite fields that takevalues in modules over noncommutative rings Λ . To each such complex we associate an L-function livingin the first K-group of the power series ring over Λ . We then show that these L-functions satisfy a suitablygeneralised multiplicative Grothendieck trace formula.

Key words: MSCs 14G10 (11G25 14G15)

1 Introduction

Let F be an `-adic sheaf on a separated scheme X of finite type over a finite field F of characteristic differentfrom `. The L-function of F is defined as the product over all closed points x of X of the characteristicpolynomials of the geometric Frobenius automorphism Fξ at a geometric point ξ over x acting on the stalkFξ :

L(F,T ) = ∏x

det(id−Fξ T degx : Fξ )−1.

The Grothendieck trace formula relates the L-function to the action of the geometric Frobenius FF onthe `-adic cohomology groups with proper support over the base change

X = X×SpecF SpecF

of X to the algebraic closure F of F:

L(F,T ) = ∏i∈Z

det(id−FFT : Hic(X ,F))(−1)i+1

.

It was used by Grothendieck to establish the rationality and the functional equation of the zeta function ofX , both of which are parts of the Weil conjectures.

The Grothendieck trace formula may also be viewed as an equality between two elements of the first K-group of the power series ring Z`[[T ]]. Since the ring Z`[[T ]] is a semilocal commutative ring, K1(Z`[[T ]])may be identified with the group of units Z`[[T ]]× via the map induced by the determinant. For each closedpoint x of X , the Z`[[T ]]-automorphism id−Fξ T on Z`[[T ]]⊗Z`

Fξ defines a class in K1(Z`[[T ]]). Theproduct of all these classes converges in the profinite topology induced on K1(Z`[[T ]]) by the isomorphism

K1(Z`[[T ]])∼= lim←−n

K1(Z`[[T ]]/(`n,T n)).

Malte WitteUniversität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, e-mail: [email protected]

311

[email protected]

[email protected]

312 Malte Witte

The image of the limit under the determinant map agrees with the inverse of the L-function of F. On theother hand, the Z`[[T ]]-automorphisms

Z`[[T ]]⊗Z`Hi

c(X ,F)id−FFT−−−−→ Z`[[T ]]⊗Z`

Hic(X ,F)

also give rise to elements in the group K1(Z`[[T ]]). The Grothendieck trace formula may thus be translatedinto an equality between the alternating product of those elements and the class corresponding to the L-function.

In this article, we will show that in the above reformulation of the Grothendieck trace formula, one mayreplace Z` by any adic Z`-algebra, i. e. a compact, semilocal Z`-algebra Λ whose Jacobson radical is finitelygenerated. These rings play an important role in noncommutative Iwasawa theory.

A central step in this reformation is the development of a convenient framework, in which one canput the K-theoretic machinery to use. This was accomplished in [Wit08], using the notion of Waldhausencategories. For any adic ring Λ , we introduced in loc. cit. a Waldhausen category of perfect complexes ofadic sheaves of Λ -modules on X . Furthermore, we presented an explicit construction of a Waldhausen exactfunctor RΓc(X ,F•) that computes the cohomology with proper support for any perfect complex F•.

By suitably adapting the classical construction, we define the L-function of such a complex F• as anelement L(F•,T ) of K1(Λ [[T ]]). The automorphism id−FFT on Λ [[T ]]⊗Λ RΓc(X ,F•) gives rise to anotherclass in K1(Λ [[T ]]). Below, we shall prove the following theorem.

Theorem 1. Let F• be a perfect complex of adic sheaves of Λ -modules on X. Then

L(F•,T ) =[Λ [[T ]]⊗Λ RΓc(X ,F•)

id−FFT−−−−→Λ [[T ]]⊗Λ RΓc(X ,F•)]−1

in K1(Λ [[T ]]).

The rough line of argumentation in the proof is as follows. As in the proof of the classical Grothendiecktrace formula, one may reduce everything to the case of X being a smooth geometrically connected curveover the finite field F. Moreover, one can replace Λ by Z`[Gal(L/K)], where L is a Galois extension of thefunction field K of X . By the classical Grothendieck trace formula, we know that our theorem is true if wefurther enlarge Z`[Gal(L/K)] to the maximal order M in a split semisimple algebra. The crucial step is thento show that

SK1(Z`[Gal(L/K)][[T ]]) = kerK1(Z`[Gal(L/K)][[T ]])→ K1(M[[T ]])

vanishes in the limit as L tends to the separable closure of K. This is achieved as follows: By using resultsof [Oli88] we prove that

SK1(OF [G][[T ]]) = SK1(OF [G])

for any finite group G and the valuation ring OF of any finite extension F of Q`. (This fact also has someother interesting applications, see e. g. [CPT12].) The vanishing of

lim←−L

SK1(OF [Gal(L/K)])

then follows by using an argument from [FK06].Outline. Section 2 recalls briefly Waldhausen’s construction of algebraic K-theory. In Section 3 we in-

troduce a special Waldhausen category that computes the K-theory of an adic ring. A similar construction isthen used in Section 4 to define the categories of perfect complexes of adic sheaves. In Section 5 we provethe abovementioned results for SK1(Z`[G][[T ]]). In Section 6 we define the L-function of a perfect complexof adic sheaves. Section 7 contains the proof of the Grothendieck trace formula for these L-functions.

Acknowledgements. The author would like to thank Annette Huber and Alexander Schmidt for theirencouragement and for valuable discussions. He also thanks the anonymous referee for various corrections.

Noncommutative L-functions for varieties over finite fields 313

2 Waldhausen Categories

Waldhausen [Wal85] introduced a construction of algebraic K-theory that is both more transparent and moreflexible than Quillen’s original approach. He associates K-groups to any category of the following kind.

Definition 1. A Waldhausen category W is a category with a zero object ∗, together with two subcategoriesco(W) (cofibrations) and w(W) (weak equivalences) subject to the following set of axioms.

1. Any isomorphism in W is a morphism in co(W) and w(W).2. For every object A in W, the unique map ∗→ A is in co(W).3. If A→ B is a map in co(W) and A→ C is a map in W, then the pushout B∪A C exists and the canonical

map C→ B∪A C is in co(W).4. If in the commutative diagram

B

Af//oo

C

B′ A′ g

//oo C′

the morphisms f and g are cofibrations and the downwards pointing arrows are weak equivalences, then thenatural map B∪A C→ B′∪A′C′ is a weak equivalence.

We denote maps from A to B in co(W) by A B, those in w(W) by A ∼−→ B. If C = B∪A ∗ is a cokernelof the cofibration A B, then we denote the natural quotient map from B to C by BC. The sequence

A BC

is called exact sequence or cofibre sequence.

Definition 2. A functor between Waldhausen categories is called (Waldhausen) exact if it preserves cofi-brations, weak equivalences, and pushouts along cofibrations.

If W is a Waldhausen category, then Waldhausen’s S-construction yields a topological space K(W) andWaldhausen exact functors F : W→W′ yield continuous maps K(F) : K(W)→K(W′) [Wal85].

Definition 3. The n-th K-group of W is defined to be the n-th homotopy group of K(W):

Kn(W) = πn(K(W)).

Example 1.

1. Any exact category E may be viewed as a Waldhausen category by taking the admissible monomorphismsas cofibrations and isomorphisms as weak equivalences. Then the Waldhausen K-groups of E agree withthe Quillen K-groups of E [TT90, Theorem 1.11.2].

2. Let Komb(E) be the category of bounded complexes over the exact category E with degreewise admissi-ble monomorphisms as cofibrations and quasi-isomorphisms (in the category of complexes of an ambientabelian category A) as weak equivalences. By the Gillet-Waldhausen theorem [TT90, Theorem 1.11.7], theWaldhausen K-groups of Komb(E) also agree with the K-groups of E .

3. In fact, Thomason showed that if W is any sufficiently nice Waldhausen category of complexes and F : W→Komb(E) a Waldhausen exact functor that induces an equivalence of the derived categories of W andKomb(E), then F induces an isomorphism of the corresponding K-groups [TT90, Theorem 1.9.8].

Remark 1. In the view of Example 1.(3) one might wonder wether it is possible to define a reasonableK-theory for triangulated categories. However, [Sch02] shows that such a construction fails to exist.

The zeroth K-group of a Waldhausen category can be described fairly explicitly as follows.

Proposition 1. Let W be a Waldhausen category. The group K0(W) is the abelian group generated by theobjects of W modulo the relations

1. [A] = [B] if there exists a weak equivalence A ∼−→ B,

314 Malte Witte

2. [B] = [A][C] if there exists a cofibre sequence A BC.

Proof. See [TT90, §1.5.6]. ut

There also exists a description of K1(W) for general W as the kernel of a certain group homomorphism[MT07]. We shall come back to this description later in a more specific situation.

3 The K-Theory of Adic Rings

All rings will be associative with unity, but not necessarily commutative. For any ring R, we let

Jac(R) = x ∈ R|1− rx is invertible for any r ∈ R

denote the Jacobson radical of R, i. e. the intersection of all maximal left ideals. It is the largest two-sidedideal I of R such that 1+ I ⊂ R× [Lam91, Chapter 2, §4]. The ring R is called semilocal if R/Jac(R) isartinian.

Definition 4. A ring Λ is called an adic ring if it satisfies any of the following equivalent conditions:

1. Λ is compact, semilocal and the Jacobson radical is finitely generated.2. For each integer n≥ 1, the ideal Jac(Λ)n is of finite index in Λ and

Λ = lim←−n

Λ/Jac(Λ)n.

3. There exists a twosided ideal I such that for each integer n≥ 1, the ideal In is of finite index in Λ and

Λ = lim←−n

Λ/In.

Example 2. The following rings are adic rings:

1. any finite ring,2. Z`,3. the group ring Λ [G] for any finite group G and any adic ring Λ ,4. the power series ring Λ [[T ]] for any adic ring Λ and an indeterminate T that commutes with all elements of

Λ ,5. the profinite group ring Λ [[G]], when Λ is a adic Z`-algbra and G is a topologically finitely generated

profinite group whose `-Sylow subgroup has finite index in G.

Note that adic rings are not noetherian in general, the power series over Z` in two noncommuting indeter-minates being a counterexample.

We will now examine the K-theory of Λ .

Definition 5. Let R be any ring. A complex M• of left R-modules is called strictly perfect if it is strictlybounded and for every n, the module Mn is finitely generated and projective. We let SP(R) denote theWaldhausen category of strictly perfect complexes, with quasi-isomorphisms as weak equivalences andinjective complex morphisms as cofibrations.

Definition 6. Let R and S be two rings. We denote by Rop-SP(S) the Waldhausen category of complexes ofS-R-bimodules (with S acting from the left, R acting from the right) which are strictly perfect as complexesof S-modules. The weak equivalences are given by quasi-isomorphisms, the cofibrations are the injectivecomplex morphisms.

By Example 1 we know that the Waldhausen K-theory of SP(R) coincides with the Quillen K-theory ofR:

Kn(SP(R)) = Kn(R).

For complexes M• and N• of right and left R-modules, respectively, we let


(M⊗R N)•

denote the total complex of the bicomplex M•⊗R N•. Any complex M• in Rop-SP(S) clearly gives rise to aWaldhausen exact functor

(M⊗R (−))• : SP(R)→ SP(S).

and hence, to homomorphisms Kn(R)→ Kn(S).Let now Λ be an adic ring. The first algebraic K-group of Λ has the following useful property.

Proposition 2 ([FK06], Prop. 1.5.3). Let Λ be an adic ring. Then

K1(Λ) = lim←−I∈IΛ

K1(Λ/I)

In particular, K1(Λ) is a profinite group.

It will be convenient to introduce another Waldhausen category that computes the K-theory of Λ .

Definition 7. Let R be any ring. A complex M• of left R-modules is called DG-flat if every module Mn

is flat (but not necessarily finitely generated) and for every acyclic complex N• of right R-modules, thecomplex (N⊗R M)• is acyclic.

Remark 2. The notion of DG-flatness goes back to Avramov and Foxby [AF91]. Every bounded abovecomplex of flat modules is DG-flat, but a DG-flat complex does not need to be bounded above and not everyunbounded complex of flat modules is DG-flat. We refer to the cited reference for examples. Unboundedcomplexes will appear quite naturally in later constructions. If one desires, one can avoid the homologicalalgebra of unbounded complexes by using appropriate truncation operations, but the author feels that theconcept of DG-flatness is the more elegant solution. See also Remark 5 in this regard.

We shall denote the lattice of open ideals of an adic ring Λ by IΛ .

Definition 8. Let Λ be an adic ring. We denote by PDGcont(Λ) the following Waldhausen category. Theobjects of PDGcont(Λ) are inverse system (P•I )I∈IΛ

satifying the following conditions:

1. for each I ∈ IΛ , P•I is a DG-flat complex of left Λ/I-modules and perfect, i. e. quasi-isomorphic to a complexin SP(Λ/I),

2. for each I ⊂ J ∈ IΛ , the transition morphism of the system

ϕIJ : P•I → P•J

induces an isomorphismΛ/J⊗Λ/I P•I ∼= P•J .

A morphism of inverse systems ( fI : P•I → Q•I )I∈IΛin PDGcont(Λ) is a weak equivalence if every fI is a

quasi-isomorphism. It is a cofibration if every fI is injective and the system (coker fI)I∈IΛis in PDGcont(Λ).

To see that PDGcont(Λ) is indeed a Waldhausen category one uses that the category PDGcont(Λ) is a fullsubcategory of the category of complexes over the abelian category of IΛ -systems of Λ -modules which isclosed under shifts and extensions; see [Wit08, Prop. 5.4.5] for a detailed proof.

Proposition 3. The Waldhausen exact functor

SP(Λ)→ PDGcont(Λ), P•→ (Λ/I⊗Λ P•)I∈IΛ

identifies SP(Λ) with a full Waldhausen subcategory of PDGcont(Λ). Moreover, it induces isomorphisms

Kn(SP(Λ))∼= Kn(PDGcont(Λ)).

Proof. The main step is to show that for every object (Q•I )I∈IΛin PDGcont(Λ), the complex

lim←−I∈IΛ

Q•I

316 Malte Witte

is a perfect complex of Λ -modules. This is proved using the argument of [FK06, Proposition 1.6.5]. Theassertion about the K-theory is then an easy consequence of the Waldhausen approximation theorem. Werefer to [Wit08, Proposition 5.2.5] for the details. ut

We will now extend the definition of the tensor product to PDGcont(Λ).

Definition 9. For (P•I )I∈IΛ∈ PDGcont(Λ) and M• ∈Λ op-SP(Λ ′) we set

ΨM((P•I )I∈IΛ

)= ( lim←−

J∈IΛ

Λ′/I⊗Λ ′ (M⊗Λ PJ)

•)I∈IΛ ′

and obtain a Waldhausen exact functor

ΨM : PDGcont(Λ)→ PDGcont(Λ ′).

Note that the annihilatorA =

x ∈Λ | Λ

′/I⊗Λ ′ M•x = 0

is an open two-sided ideal of Λ and therefore, an element of IΛ . If J ∈ IΛ is contained in this annihilator,then we have


• ∼= (Λ ′/I⊗Λ ′ M⊗Λ/A Λ/A⊗Λ/J PJ)• ∼= Λ

′/I⊗Λ ′ (M⊗Λ PA)•.

In particular, since IΛ is filtered,

lim←−J∈IΛ


• ∼= Λ′/I⊗Λ ′ (M⊗Λ PA)

•.

With this description, it is clear that ΨM((F•I )I∈IΛ

)is indeed an object in the category PDGcont(X ,Λ ′). We

refer to [Wit08, Prop. 5.5.4] for the easy proof that ΨM is Waldhausen exact.

Proposition 4. Let M• be a complex in Λ op-SP(Λ ′). Then the following diagram commutes.

Kn(SP(Λ))∼= //

Kn(M•⊗Λ (−))

Kn(PDGcont(Λ))

Kn(ΨM)

Kn(SP(Λ ′))

∼= // Kn(PDGcont(Λ ′))

Proof. Let P• be a strictly perfect complex in SP(Λ). There exists a canonical isomorphism(Λ′/I⊗Λ ′ (M⊗Λ P)•

)I∈I

Λ ′∼= ( lim←−

J∈IΛ

Λ′/I⊗Λ ′ (M⊗Λ Λ/J⊗Λ P)•)I∈I

Λ ′.

ut

From [MT07] we deduce the following properties of the group K1(Λ).

Proposition 5. The group K1(Λ) is generated by the weak autoequivalences

( fI : P•I∼−→ P•I )I∈IΛ

in PDGcont(Λ). Moreover, we have the following relations:

1. [( fI : P•I∼−→ P•I )I∈IΛ

] = [(gI : P•I∼−→ P•I )I∈IΛ

][(hI : P•I∼−→ P•I )I∈IΛ

] if for each I ∈ IΛ , one has fI = gI hI ,2. [( fI : P•I

∼−→ P•I )I∈IΛ] = [(gI : Q•I

∼−→Q•I )I∈IΛ] if for each I ∈ IΛ , there exists a quasi-isomorphism aI : P•I

∼−→Q•I such that the square

P•IfI //

aI

P•I

aI

Q•I

gI // Q•I


commutes up to homotopy,3. [(gI : P′•I

∼−→ P′•I )I∈IΛ] = [( fI : P•I

∼−→ P•I )I∈IΛ][(hI : P′′•I

∼−→ P′′•I )I∈IΛ] if for each I ∈ IΛ , there exists an exact

sequence PI P′I P′′I such that the diagram

P•I

fI

// // P′•I

gI

// // P′′•I

hI

P•I // // P′•I // // P′′•I

commutes in the strict sense.

Proof. The description of K1(PDGcont(Λ)) as the kernel of

D1PDGcont(Λ)∂−→D0PDGcont(Λ)

given in [MT07] shows that the weak autoequivalences are indeed elements of K1(PDGcont(Λ)). Togetherwith Proposition 2 this description also implies that relations (1) and (3) are satisfied. For relation (2), onecan use [Wit08, Lemma 3.1.6]. Finally, the classical description of K1(Λ) implies that K1(PDGcont(Λ))is already generated by isomorphisms of finitely generated, projective modules viewed as strictly perfectcomplexes concentrated in degree 0. ut

Remark 3. Despite the relatively explicit description of K1(W) for a Waldhausen category W in [MT07] itis not an easy task to deduce from it a presentation of K1(W) as an abelian group. We refer to [MT08] for apartial result in this direction.

In particular, one should not expect that the relations (1)–(3) describe the group K1(PDGcont(Λ)) com-pletely. However, they will suffice for the purpose of this paper.

4 Perfect Complexes of Adic Sheaves

We let F denote a finite field of characteristic p, with q = pν elements. Furthermore, we fix an algebraicclosure F of F.

Write SchsepF for the category of separated F-schemes of finite type. For any scheme X in Schsep

F and anyadic ring Λ we introduced in [Wit08] a Waldhausen category PDGcont(X ,Λ) of perfect complexes of adicsheaves on X . Below, we will recall the definition.

Definition 10. Let R be a finite ring and X be a scheme in SchsepF . A complex F• of étale sheaves of left

R-modules on X is called strictly perfect if it is strictly bounded and each Fn is constructible and flat. Acomplex is called perfect if it is quasi-isomorphic to a strictly perfect complex. It is DG-flat if for eachgeometric point of X , the complex of stalks is DG-flat.

Definition 11. We will denote by PDG(X ,R) the category of DG-flat perfect complexes of R-modules onX . It is a Waldhausen category with quasi-isomorphisms as weak equivalences and injective complex mor-phisms with cokernel in PDG(X ,R) as cofibrations.

Definition 12. Let X be a scheme in SchsepF and let Λ be an adic ring. The category of perfect complexes

of adic sheaves PDGcont(X ,Λ) is the following Waldhausen category. The objects of PDGcont(X ,Λ) areinverse system (F•I )I∈IΛ

such that:

1. for each I ∈ IΛ , F•I is in PDG(X ,Λ/I),2. for each I ⊂ J ∈ IΛ , the transition morphism

ϕIJ : F•I → F•J

of the system induces an isomorphismΛ/J⊗Λ/I F

•I∼= F•J .

318 Malte Witte

Weak equivalences and cofibrations are those morphisms of inverse systems that are weak equivalences orcofibrations for each I ∈ IΛ , respectively.

We refer to [Wit08, Cor. 4.1.4, Prop. 5.4.5] for the straightforward verification that PDG(X ,Λ) andPDGcont(X ,Λ) are indeed Waldhausen categories.

Remark 4. If Λ is a finite ring, the zero ideal is open and hence, an element in IΛ . In particular, the followingWaldhausen exact functors are mutually inverse equivalences for finite rings Λ :

PDGcont(X ,Λ)→ PDG(X ,Λ), (F•I )I∈IΛ7→ F•(0),

PDG(X ,Λ)→ PDGcont(X ,Λ), F• 7→ (Λ/I⊗Λ F•)I∈IΛ.

We use these functors to identify the two categories.

If Λ = Z`, then the subcategory of complexes concentrated in degree 0 of PDGcont(X ,Z`) correspondsprecisely to the exact category of those constructible `-adic sheaves on X in the sense of [Gro77, Exp. VI,Def. 1.1.1, Exp. V, Def. 3.1.1] which are flat. In this sense, we recover the classical theory.

If f : Y → X is a morphism in SchsepF , we define a Waldhausen exact functor

f ∗ : PDGcont(X ,Λ)→ PDGcont(Y,Λ), (F•I )I∈IΛ7→ ( f ∗F•I )I∈IΛ

.

We will also need a Waldhausen exact functor that computes higher direct images with proper support. Forthe purposes of this article it suffices to use the following construction.

For any X in SchsepF and any complex G• of abelian étale sheaves on X we let G•X G denote the Godement

resolution of G• [Wit08, Def. 4.2.1]. We note that G•X G is a complex of flabby sheaves quasi-isomorphicto G• [Wit08, Lemma 4.2.3, Prop. 4.26]. Moreover, for any finite ring Λ the Godement resolution is aWaldhausen exact functor

G•X : PDG(X ,Λ)→ PDG(X ,Λ)

[Wit08, Cor. 4.2.8].

Remark 5. We point out that with the above definition, G•X G is in general not bounded above, even if G•

is concentrated in degree 0 and that GnX G is not a constructible sheaf of Λ -modules, even if this is true

for all Gn, n ∈ Z. Since X is of finite Krull dimension, one can also work with a truncated version of theGodement resolution that preserves boundedness of complexes, but we do not know of any resolution withgood functorial properties taking strictly perfect complexes to strictly perfect complexes.

By a theorem of Nagata [Nag63, Thm. 2] there exist a factorisation f = p j for any morphism f : X→Yin the category Schsep

F of separated schemes of finite type over F such that j : X → X ′ is an open immersionand p : X ′→ Y is a proper morphism.

Definition 13. Define a Waldhausen exact functor

R f! : PDGcont(X ,Λ)→ PDGcont(Y,Λ)

(F•I )I∈IΛ7→ (p∗G•X ′ j!FI)I∈IΛ

For the verification that the definition of R f! makes sense and produces a Waldhausen exact functor werefer to [Wit08, Prop. 4.3.4, Prop. 4.3.8, Def. 5.4.13].

Obviously, this definition depends on the particular choice of the compactification f = p j. However,all possible choices will induce the same homomorphisms

Kn(R f!) : Kn(PDGcont(X ,Λ))→ Kn(PDGcont(Y,Λ))

and this is all we need.

Remark 6. In [Wit08, Section 4.5] we present a way to make the construction of R f! independent of thechoice of a particular compactification.

Proposition 6. Let f : X → Y be a morphism in SchsepF .


1. Kn(R f!) is independent of the choice of a compactification f = p j.2. Let F′ be a subfield of F and consider f as a morphism in Schsep

F′ . Then Kn(R f!) remains the same.3. If g : Y → Z is another morphism in Schsep

F , then

Kn(R(g f )!) = Kn(Rg!)Kn(R f!)

4. For any cartesian square

Y ×X Zf ′ //

g′

Z

g

Yf // X

in SchsepF we have

Kn( f ∗Rg!) = Kn(Rg′! f ′∗)

Proof. All of this follows easily from [AGV72, Exposé XXVII]. See also [Wit08, Section 4.5]. ut

Definition 14. Let X be a scheme in SchsepF and write h : X → SpecF for the structure map, s : SpecF→

SpecF for the map induced by the embedding into the algebraic closure. We define the Waldhausen exactfunctor

RFΓc(X ,−) : PDGcont(X ,Λ)→ PDGcont(Λ)

to be the composition ofRh! : PDGcont(X ,Λ)→ PDGcont(SpecF,Λ)

with the section functor

PDGcont(SpecF,Λ)→ PDGcont(Λ), (F•I )I∈IΛ→ (Γ (SpecF,s∗F•I ))I∈IΛ

.

Remark 7. If F′ is a subfield of F, then RFΓc(X ,−) and RF′ Γc(X ,−) are in fact quasi-isomorphic andhence, they induce the same homomorphism of K-groups. Nevertheless, it will be convenient to distinguishbetween the two functors. We will omit the index if the base field is clear from the context.

The definition of ΨM extends to PDGcont(X ,Λ).

Definition 15. For (F•I )I∈IΛ∈ PDGcont(X ,Λ) and M• ∈Λ op-SP(Λ ′) we set

ΨM((F•I )I∈IΛ

)= ( lim←−

J∈IΛ

Λ′/I⊗Λ ′ (M⊗Λ FJ)

•)I∈IΛ ′

and obtain a Waldhausen exact functor

ΨM : PDGcont(X ,Λ)→ PDGcont(X ,Λ ′).

Proposition 7. Let M• be a complex in Λ op-SP(Λ ′). Then the natural morphism

ΨM RΓc(X ,−)→ RΓc(X ,ΨM(−))

is a weak equivalence. In particular, the following diagram commutes.

Kn(PDGcont(X ,Λ))Kn RΓc(X ,−) //

Kn(ΨM)

Kn(PDGcont(Λ))

Kn(ΨM)

Kn(PDGcont(X ,Λ ′))

Kn RΓc(X ,−) // Kn(PDGcont(Λ ′))

Proof. See [Wit08, Proposition 5.5.7]. ut

Finally, we need the following result. Let X be a connected scheme and f : Y →X a finite Galois coveringof X with Galois group G , i. e. f is finite étale and the degree of f is equal to the order of G = AutX (Y ).We set

320 Malte Witte

Z`[G]]X = (Z`[G]/I⊗Z[G] f! f ∗Z)I∈IZ`[G].

Then Z`[G]]X is a system of locally constant sheaves which we may view as an object of PDGcont(X ,Z`[G])concentrated in degree 0. It corresponds to the continuous Galois module Z`[G] on which the fundamentalgroup of X acts contragrediently.

Lemma 1. Let X be a connected scheme in SchsepF . Let R be a finite Z`-algebra and let F• be a bounded

complex of flat, locally constant, and constructible sheaves of R-modules. Then there exists a finite Galoiscovering Y of X with Galois group G and a complex M• in Z`[G]op-SP(R) such that

F• ∼=ΨM(Z`[G]]X )

in PDG(X ,R).

Proof. Choose a large enough Galois covering f : Y → X such that f ∗F• is a complex of constant sheavesand set M• = Γ (Y, f ∗F•). This is in a natural way a complex in Z`[G]op-SP(R) and

M•⊗Z`[G]Z`[G]]X∼= F•

See [Wit08, Section 5.6] for further details. ut

5 On SK1(Z`[G][[T ]])

In this section, we will fix a finite extension F of Q` and write OF for the valuation ring of F . Let G be afinite group and let T denote an indeterminate that commutes with every element of OF [G]. In the end, wewill only need the case F =Q`, but the more general results of this section may be interesting also for otherapplications.

Recall that there exists a finite extension F ′ of F such that F ′[G] is split semisimiple:

F ′[G]∼=r

∏k=1

EndopF ′(F

′sk)

for some integers r,s1, . . . ,sr. Let M be a maximal OF -order in F ′[G], i. e. an OF -lattice in F ′[G] which is asubring and which is maximal with respect to this property. Then

M ∼=r

∏k=1

EndopOF ′

(OskF ′)

according to [Oli88, Theorem 1.9]. In particular, the determinant map induces an isomorphism

K1(M)∼=r⊕

k=1

O×F ′ .

If OF [G]⊂M, then Theorem 2.5 of loc. cit. implies

SK1(OF [G]) = kerK1(OF [G])→ K1(F [G]) = kerK1(OF [G])→ K1(M).

Analogously, we define a subgroup in K1(OF [G][[T ]]).

Definition 16. Let G be a finite group and choose a finite extension F ′ of F such that F ′[G] is split semisim-ple. We set

SK1(OF [G][[T ]]) = kerK1(OF [G][[T ]])→ K1(M[[T ]])

where M denotes a maximal OF -order in F ′[G] containing OF [G].

Remark 8. Let Q be an algebraic closure of the fraction field of OF [[T ]]. Then

SK1(OF [G][[T ]]) = kerK1(OF [G][[T ]])→ K1(Q[G])


such that our definition agrees with the definition given in [CPT13].

Lemma 2. For any finite group G,

SK1(OF [G][[T ]])∼= lim←−n

SK1(OF [G×Z/(`n)]).

Proof. Let F ′ and F ′′ be splitting fields for OF [G] and OF [G×Z/(`n)], respectively and denote the corre-sponding maximal orders by M and M′. The commutativity of the diagram

K1(M[Z/(`n)]) //

∼= det

K1(M′)

∼= detr⊕

k=1OF ′ [Z/(`n)]×

⊂ //r′⊕

k=1O×F ′′

implies thatSK1(OF [G×Z/(`n)]) = kerK1(OF [G×Z/(`n)])→ K1(M[Z/(`n)]).

By [NSW00, Theorem 5.3.5] the choice of a topological generator γ ∈ Z` induces an isomorphism

OF [[T ]]∼= lim←−nOF [Z/(`n)], T 7→ γ−1.

In particular, we have compatible isomorphisms

K1(OF [G][[T ]])∼= lim←−n

K1(OF [G×Z/(`n)]), K1(M[[T ]])∼= lim←−n

K1(M[Z/(`n)])

by Proposition 2. Hence, we obtain an isomorphism

SK1(OF [G][[T ]])∼= lim←−n

SK1(OF [G×Z/(`n)])

as claimed. ut

Proposition 8. For any finite group G,

SK1(OF [G][[T ]])∼= SK1(OF [G]).

Proof. By Lemma 2 it suffices to prove that the projection OF [G×Z/(`n)]→ OF [G] induces an isomor-phism

SK1(OF [G×Z/(`n)])∼= SK1(OF [G]).

Let g1, . . . ,gk be a system of representatives for the F-conjugacy classes of elements of order prime to` in G. (Two elements g, h of order r prime to ` are called F-conjugated if ga = xhx−1 for some x ∈ G,a ∈ Gal(F(ζr)/F)⊂ (Z/(r))×.) Let ri denote the order of gi and set

Ni(G) =

x ∈ G | xgix−1 = gai for some a ∈ Gal(F(ζri)/F)

,

Zi(G) =

x ∈ G | xgix−1 = gi.

Furthermore, let

Hab2 (Zi(G),Z) = Im

⊕H⊂Zi(G)

abelian

H2(H,Z)→ H2(Zi(G),Z)

denote the subgroup of the second homology group generated by elements induced up from abelian sub-groups of Zi(G). According to [Oli88, Theorem 12.5] there exists an isomorphism

322 Malte Witte

SK1(OF [G])∼=k⊕

i=1

H0(Ni(G)/Zi(G),H2(Zi(G),Z)/Hab2 (Zi(G),Z))(`).

Now, (g1,0), . . . ,(gk,0) is a system of representatives for the F-conjugacy classes of elements of orderprime to ` in G×Z/(`n) and

Ni(G×Z/(`n)) = Ni(G)×Z/(`n), Zi(G×Z/(`n)) = Zi(G)×Z/(`n).

By [Oli88, Prop. 8.12], we have

H2(Zi(G)×Z/(`n),Z)/Hab2 (Zi(G)×Z/(`)n,Z) =

H2(Zi(G),Z)/Hab2 (Zi(G),Z)×H2(Z/(`n),Z)/Hab

2 (Z/(`n),Z)

and clearly,H2(Z/(`n),Z) = Hab

2 (Z/(`n),Z).

From this, the claim of the proposition follows. ut

Remark 9. In the case that F is unramified over Q` the above equality was independently observed byChinburg, Pappas, and Taylor [CPT13] (using a different approach) some years after the first version of thisarticle had been made available.

The following proposition was proved in [FK06, Proposition 2.3.7] in the case of number fields andF =Q`.

Proposition 9. Let Q be a function field of transcendence degree 1 over a finite field F and let ` be anyprime. Then

lim←−L

SK1(OF [Gal(L/Q)]) = 0.

where L runs through the finite Galois extensions of Q in a fixed separable closure Q of Q.

Proof. Let F ′ be a totally ramified extension of F . Using [Oli88, Theorem 8.7.(i)] and the functorialityof the construction of the isomorphism in [Oli88, Theorem 12.5] one checks that the inclusion OF → OF ′

induces an isomorphismSK1(OF [G])→ SK1(OF ′ [G])

for any finite group G. Hence, we may assume that F is unramified over Q`. We then have a surjection

H2(G,OF [Gr])→ SK1(OF [G])

with OF [Gr] denoting the OF -module generated by the elements in G of order prime to ` [Oli88, Theorem12.10].

By the same argument as in the proof of Proposition 2.3.7 in [FK06], it now suffices to prove that

H2(Gal(Q/L),F/OF) = 0

for any finite extension L of Q.If ` is different from the characteristic of F, then the vanishing of this group can be deduced via the same

argument as the analogous statement for number fields given in [Sch79, § 4, Satz 1]: Let

L∞ =⋃n

L(ζ`n).

ThenH2(Gal(Q/L),F/OF) = H1(Gal(L∞/L),H1(Gal(Q/L∞),F/OF))

and by Kummer theory,H1(Gal(Q/L∞),F/OF) = L×∞ ⊗Z F/OF(−1).

Now


H1(Gal(L∞/L),L×∞ ⊗Z F/OF(−1)) = lim−→n

H1(Gal(L∞/L),L(ζ`n)×⊗Z F/OF(−1))

by [NSW00, Proposition 1.5.1] and

H1(Gal(L∞/L),L(ζ`n)×⊗Z F/OF(−1)) =(L(ζ`n)×⊗Z F/OF(−1)

)Gal(L∞/L)

by loc. cit., Proposition 1.6.13. Since the latter group is a factor group of(L(ζ`n)×⊗Z F/OF(−1)

)Gal(L∞/L(ζ`n ))

= 0,

the claim is proved.If ` is equal to the characteristic of F, then the cohomological `-dimension of Gal(Q/L) is known to be

1 [NSW00, Theorem 10.1.11.(iv)] and hence, the second cohomology group of F/OF vanishes for trivialreasons. ut

6 L-Functions

Consider an adic ring Λ and let Λ [[T ]] denote the ring of power series in the indeterminate T (where T isassumed to commute with every element of Λ ). The ring Λ [[T ]] is still an adic ring whose Jacobson radicalJac(Λ [[T ]]) is generated by Jac(Λ) and T . In particular, we conclude from Proposition 2 that

K1(Λ [[T ]]) = lim←−n

K1(Λ [[T ]]/Jac(Λ [[T ]])n)

is a profinite group.Let F be a finite field. We write X0 for the set of closed points of a scheme X in Schsep

F . If x ∈ X0 is aclosed point, then

x = x×SpecF SpecF

consists of finitely many points, whose number is given by the degree deg(x) of x, i. e. the degree of theresidue field k(x) of x as a field extension of F. Let

sx : x→ X

denote the structure map. For any complex

F• = (F•I )I∈IΛ

in PDGcont(X ,Λ), we writeF•x = (Γ (x,s∗xF

•I ))I∈IΛ

.

This defines a Waldhausen exact functor

PDGcont(X ,Λ)→ PDGcont(Λ), F• 7→ F•x .

Note that F•x can also be written as the product over the stalks of F in the points of x:

F•x∼= ∏

ξ∈x((F•I )ξ )I∈Λ .

The geometric Frobenius automorphism

FF ∈ Gal(F/F)

operates on F•x through its action on x. Hence, it also operates on ΨΛ [[T ]](F•x ). Here,

ΨΛ [[T ]] : PDGcont(Λ)→ PDGcont(Λ [[T ]])

324 Malte Witte

denotes the change of ring functor with respect to the Λ [[T ]]-Λ -bimodule Λ [[T ]], as constructed in Defini-tion 9. The morphism

id−FFT : ΨΛ [[T ]](F•x )→ΨΛ [[T ]](F

•x ).

is a natural isomorphism whose inverse is given by

∞

∑n=0

FnFT n.

Definition 17. The class

Ex(F•,T ) = [ΨΛ [[T ]](F

•x )

id−FFT−−−−→ΨΛ [[T ]](F•x )]−1

in K1(Λ [[T ]]) is called the Euler factor of F• at x.

One can easily verify that the Euler factor is multiplicative on exact sequences and that

Ex(F•,T ) = Ex(G

•,T )

if the complexes F• and G• are quasi-isomorphic. Hence, the above assignment extends to a homomorphism

Ex(−,T ) : K0(PDGcont(X ,Λ))→ K1(Λ [[T ]]).

Lemma 3. Let ξ ∈ x be a geometric point. Then

Ex(F•,T ) = [ΨΛ [[T ]](F

•ξ)

id−Fk(x)Tdeg(x)

−−−−−−−−−→ΨΛ [[T ]](F•ξ)]−1.

Proof. The Frobenius automorphism FF induces isomorphisms F•FkFξ

∼= F•ξ

for k = 1, . . . ,deg(x). For k =

deg(x) we have FkFξ = ξ and the isomorphism is given by the operation of Fk(x) on F•

ξ. Hence, we may

identify F•x with the complex (F•ξ)deg(x), on which the Frobenius FF acts through the matrix

0 . . . . . 0 Fk(x)id 0 . . . . . 00 id 0 . . . 0...

. . . . . . . . ....

0 . . . 0 id 0

.

Let A be the automorphism of ΨΛ [[T ]]((F•ξ)deg(x)) given by the matrix

id 0 . . . . . . . . 0idT id 0 . . . 0

idT 2 idT id 0...

.... . . . . . . . . 0

idT deg(x)−1 . . . idT 2 idT id

Then A(id−FFT ) corresponds to the matrix

id 0 . . . 0 −Fk(x)T0 id 0 . . . −Fk(x)T 2

.... . .

...0 . . . 0 id −Fk(x)T deg(x)−1

0 . . . . . 0(id−Fk(x)T deg(x)

)

Moreover, we have [A] = 1 in K1(Λ [[T ]]). Hence,


[ΨΛ [[T ]](F•x )

id−FFT−−−−→ΨΛ [[T ]](F•x )] = [ΨΛ [[T ]](F

•ξ)

id−Fk(x)Tdeg(x)

−−−−−−−−−→ΨΛ [[T ]](F•ξ)]

as claimed. ut

Proposition 10. The infinite product∏

x∈X0

Ex(F•,T )

converges in the profinite topology of K1(Λ [[T ]]).

Proof. For each integer m, there exist only finitely many closed points x ∈ X0 with deg(x)< m. If deg(x)≥m, then we conclude from Lemma 3 that the image of Ex(F

•,T ) in K1(Λ [T ]/(T m)) is 1. ut

Definition 18. The L-function of the complex F• in PDGcont(X ,Λ) is given by

LF(F•,T ) = ∏

x∈X0

Ex(F•,T ) ∈ K1(Λ [[T ]])

Remark 10. If F′ is a subfield of F, then Lemma 3 implies that

LF′(F•,T ) = LF(F

•,T [F:F′]) ∈ K1(Λ [[T ]]).

Remark 11. If Λ is commutative, the determinant induces an isomorphism

det : K1(Λ [[T ]])→Λ [[T ]]×.

In particular, we see that the L-function agrees with the one defined in [Del77, Fonction L mod `n] in thecase of commutative adic rings.

7 The Grothendieck trace formula

In this section, we will prove the Grothendieck trace formula for our L-functions.

Definition 19. For a scheme X in SchsepF and a complex F• in PDGcont(X ,Λ) we let LF(F

•,T ) denote theelement [

ΨΛ [[T ]](

RFΓc(X ,F•)) id−FFT−−−−→ΨΛ [[T ]]

(RFΓc(X ,F•)

)]−1

in K1(Λ [[T ]]).

Theorem 2 (Grothendieck trace formula). Let F be a finite field of characteristic p and let Λ be an adicring such that p is invertible in Λ . Then

LF(F•,T ) = LF(F

•,T )

for every scheme X in SchsepF and every complex F• in PDGcont(X ,Λ).

We proceed by a series of lemmas, following closely along the lines of [Mil80, Chapter VI, §13].

Lemma 4. Let U be an open subscheme of X with closed complement Z. Theorem 2 is true for X if it istrue for U and Z.

Proof. Write j : U → X and i : Z →p X for the corresponding immersions,

u : U → SpecF, x : X → SpecF, z : Z→ SpecF

for the structure morphisms. Clearly,

LF(F•,T ) = LF( j∗F•,T )LF(i∗F•,T )

326 Malte Witte

On the other hand, we have an exact sequence

Rx! j! j∗F• Rx!F• Rx!i∗i∗F•

and (chains of) quasi-isomorphisms

Ru! j∗F• ' Rx! j! j∗F• Rz!i∗F• ' Rx!i∗i∗F•.

Hence,[Rx!F

•] = [Ru! j∗F•][Rz!i∗F•] (1)

in K0 PDGcont(SpecF,Λ) by Proposition 1. Using Proposition 5 one checks that the assignment

G• 7→ [ΨΛ [[T ]](Γ (SpecF,s∗G•)) id−FFT−−−−→ΨΛ [[T ]](Γ (SpecF,s∗G•))]−1

extends to a homomorphismK0 PDGcont(SpecF,Λ)→ K1(Λ [[T ]]),

which preserves relation (1). ut

Lemma 5. Let Λ ′ be a second adic ring and M• be a complex in Λ op-SP(Λ ′). For any F• in PDGcont(X ,Λ)we have

K1(ΨM⊗Λ Λ [[T ]])(LF(F•,T )) = LF(ΨMF•,T ),

K1(ΨM⊗Λ Λ [[T ]])(LF(F•,T )) = LF(ΨMF•,T )

in PDGcont(Λ ′[[T ]]).

Proof. For LF(F•,T ), this follows from the definition and from Proposition 7. It is true for LF(F

•,T )because it is true for all Ex(F

•,T ), x ∈ X0. ut

Next, we prove that the formula is compatible with change of the base field.

Lemma 6. Let F′ be a subfield of F and X a scheme in SchsepF . Then

LF′(F•,T ) = LF(F

•,T [F:F′]).

Proof. Let r : SpecF→ SpecF′ be the morphism induced by the inclusion F′ ⊂ F and write

h : X×SpecF SpecF→ X , h′ : X×SpecF′ SpecF→ X

s : SpecF→ SpecF, s′ : SpecF→ SpecF′

for the corresponding structure morphisms. For any F• in PDGcont(X ,Λ), the complexes r∗Rh!F•, Rr! Rh!F

•,and Rh′!F

• in PDGcont(SpecF′,Λ) are quasi-isomorphic. Moreover, for any complex G• in PDGcont(SpecF,Λ),the following diagram is commutative:

Γ (SpecF,s∗r∗r∗G•)∼= //

FF′

[F:F′]⊕k=1

Γ (SpecF,s∗G•)

0 ... 0 FFid 0 ... 0...

. . . . . ....

0 ... id 0

Γ (SpecF,s∗r∗r∗G•)∼= //

[F:F′]⊕k=1

Γ (SpecF,s∗G•)

As in the proof of Lemma 3 one concludes


[ΨΛ [[T ]]

(RF′ Γc(X ,F•)

) id−FF′T−−−−−→ΨΛ [[T ]](

RF′ Γc(X ,F•))]

=[ΨΛ [[T ]]

(Γ (SpecF,s∗r∗r∗Rh!F

•)) id−FF′T−−−−−→ΨΛ [[T ]]

(Γ (SpecF,s∗r∗r∗Rh!F

•))]

=[ΨΛ [[T ]]

(Γ (SpecF,s∗Rh!F

•)) id−FFT [F:F′]−−−−−−−→ΨΛ [[T ]]

(Γ (SpecF,s∗Rh!F

•))]

=[ΨΛ [[T ]]

(RFΓc(X ,F•)

) id−FFT [F:F′]−−−−−−−→ΨΛ [[T ]]

(RFΓc(X ,F•)

)].

ut

Clearly, Theorem 2 is true for schemes of dimension 0. Next, we consider the case that X is a curve.

Lemma 7. The formula in Theorem 2 is true for any smooth and geometrically connected curve X, Λ =

Z`[G], and F• = Z`[G]]X , where ` is a prime different from the characteristic of F and G is the Galois groupof a finite Galois covering of X.

Proof. Let Q be the function field of X and let F the function field of a finite Galois covering of X , i. e.F/Q is a finite Galois extension unramified in the closed points of X . Let dF denote the element

dF = L(Z`[Gal(F/Q)]]X ,T )L(Z`[Gal(F/Q)]]X ,T )−1

in K1(Z`[Gal(F/Q)][[T ]]).Note that dF does not change if we replace X by an open subscheme of X . Hence, by shrinking X

appropriately, we may extend the definition of dF to arbitrary finite Galois extensions F of Q. If F ⊂ F ′ andF ′/Q is Galois, then dF ′ is mapped onto dF under the canonical homomorphism

K1(Z`[Gal(F ′/Q)][[T ]])→ K1(Z`[Gal(F/Q)][[T ]]).

To see this, choose X sufficiently small such that F ′/Q is unramified in the closed points of X and applyLemma 5.

Let L be a splitting field for Q`[Gal(F/Q)] and

M ⊂ L[Gal(F/Q)]

a maximal Z`-order containing OL[G]. Recall that

M =r

∏k=1

EndopOL

(Pk)

for some finitely generated free OL-modules Pk and some r > 0 [Oli88, Theorem 1.9]. We may thus considerPk as an element in Z`[G]op-SP(OL). By the classical Grothendieck trace formula [Del77, Fonction L mod`n, Theorem 2.2.(a)] applied to the OL-sheaves ΨPk(Z`[Gal(F/Q)]]X ) and by Lemma 5, the image of dFunder the homomorphism

K1(Z`[Gal(F/Q)][[T ]])→ K1(M[[T ]])∼=−→

r⊕k=1

OL[[T ]]×

is trivial; hence dF ∈ SK1(Z`[Gal(F/Q)][[T ]]) = SK1(Z`[Gal(F/Q)]). From Proposition 9 we concludedF = 1. ut

Lemma 8. The formula in Theorem 2 is true for any scheme X in SchsepF of dimension less or equal 1, any

adic ring Λ with p ∈Λ× and any complex F• in PDGcont(X ,Λ).

Proof. By Proposition 2 it suffices to consider finite rings Λ . The `-Sylow subgroups of Λ are subrings ofΛ and Λ is equal to their direct product. Since p is invertible, the p-Sylow subgroup is trivial. Hence, wemay further assume that Λ is a Z`-algebra for ` 6= p.

Shrinking X if necessary we may assume that X is smooth, irreducible curve and that F• is a strictlyperfect complex of locally constant sheaves. By replacing F with its algebraic closure in the function field

328 Malte Witte

of X and using Lemma 6, we may assume that X is geometrically connected. By Lemma 1 and Lemma 5we have

L(F•,T ) = K1(ΨM⊗Z`[G]Z`[G][[T ]])(L(Z`[G]]X ,T ))

for a suitable Galois group G and a complex M• in Z`[G]op-SP(Λ). Likewise,

L(F•,T ) = K1(ΨM⊗Z`[G]Z`[G][[T ]])(L(Z`[G]]X ,T )).

Now the assertion follows from Lemma 7. ut

We complete the proof of Theorem 2 by induction on the dimension d of X . By shrinking X if necessarywe may assume that there exists a morphism f : X → Y such that Y and all fibres of f have dimension lessthan d. Then Proposition 6.(3) and the induction hypothesis imply

L(F•,T ) = L(R f!F•,T ) = L(R f!F

•,T ).

Let now y be a closed point of Y . Write fy : Xy→ X for the fibre over y. Then

Ey(R f!F•,T ) =

[ΨΛ [[T ]]

(RΓc(Xy, f ∗y F

•)) id−FFT−−−−→ΨΛ [[T ]]

(RΓc(Xy, f ∗y F

•))]−1

= L( f ∗y F•,T )

by Proposition 6.(4) and the induction hypothesis. Since clearly

L(F•,T ) = ∏y∈Y0

L( f ∗y F•,T ),

Theorem 2 follows.

Remark 12. The formula in Theorem 2 is also valid if Λ is a finite field of characteristic p, see [Del77,Fonction L mod `n, Theorem 2.2.(b)]. However, it does not extend to general adic Zp-algebras. We refer toloc. cit., §4.5 for a counterexample. More precisely, if Λ is a commutative Zp-algebra, then the differencebetween LF(F

•,T ) and LF(F•,T ) is given by a unit in

Λ〈T 〉= lim←−I∈I(Λ)

Λ/I[T ]

[EK01]. In [Wit13] we generalise this result to noncommutative Zp-algebras Λ .

References

AF91. L. L. Avramov and H.-B. Foxby, Homological dimensions of unbounded complexes, Journal of Pure and AppliedAlgebra (1991), no. 71, 129–155.

AGV72. M. Artin, A. Grothendieck, and J.L. Verdier, Théorie des topos et cohomologie étale des schémas (SGA 4-3), LectureNotes in Mathematics, no. 305, Springer, Berlin, 1972.

CPT12. T. Chinburg, G. Pappas, and M. J. Taylor, Higher adeles and non-abelian Riemann-Roch, preprint,arXiv:1204.4520v2, 2012.

CPT13. T. Chinburg, G. Pappas, K1 of a p-adic group ring II. The determinantal kernel SK1, preprint, arXiv:1303.5337v1,2013.

Del77. P. Deligne, Cohomologie étale (SGA 4 12 ), Lecture Notes in Mathematics, no. 569, Springer, Berlin, 1977.

EK01. M. Emerton and M. Kisin, Unit L-functions and a conjecture of Katz, Ann. of Math. 153 (2001), no. 2, 329–354.FK06. T. Fukaya and K. Kato, A formulation of conjectures on p-adic zeta functions in non-commutative Iwasawa theory,

Proceedings of the St. Petersburg Mathematical Society (Providence, RI), vol. XII, Amer. Math. Soc. Transl. Ser. 2,no. 219, American Math. Soc., 2006, pp. 1–85.

Gro77. A. Grothendieck, Cohomologie `-adique et fonctions L (SGA 5), Lecture Notes in Mathematics, no. 589, Springer,Berlin, 1977.

Lam91. T. Y. Lam, A first course in noncommutative rings, Graduate Texts in Mathematics, no. 131, Springer, Berlin, 1991.Mil80. J. S. Milne, Etale cohomology, Princeton Mathematical Series, no. 33, Princeton University Press, New Jersey, 1980.MT07. F. Muro and A. Tonks, The 1-type of a Waldhausen K-theory spectrum, Advances in Mathematics 216 (2007), no. 1,

178–211.


MT08. F. Muro and A. Tonks, On K1 of a Waldhausen category, K-theory and noncommutative geometry, EMS Series ofCongress Reports, 2008, pp. 91–116.

Nag63. M. Nagata, A generalization of the imbedding problem of an abstract variety in a complete variety, J. Math. KyotoUniv. (1963), no. 3, 89–102.

NSW00. J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of number fields, Grundlehren der mathematischen Wis-senschaften, no. 323, Springer Verlag, Berlin Heidelberg, 2000.

Oli88. R. Oliver, Whitehead groups of finite groups, London Mathematical Society lecture notes series, no. 132, CambridgeUniversity Press, Cambridge, 1988.

Sch79. P. Schneider, Über gewisse Galoiscohomologiegruppen, Math. Z. 260 (1979), 181–205.Sch02. M. Schlichting, A note on K-theory and triangulated categories, Invent. Math. 150 (2002), no. 1, 111–116.TT90. R. W. Thomason and T. Trobaugh, Higher algebraic K-theory of schemes and derived categories, The Grothendieck

Festschrift, vol. III, Progr. Math., no. 88, Birkhäuser, 1990, pp. 247–435.Wal85. F. Waldhausen, Algebraic K-theory of spaces, Algebraic and Geometric Topology (Berlin Heidelberg), Lecture Notes

in Mathematics, no. 1126, Springer, 1985, pp. 318–419.Wit08. M. Witte, Noncommutative Iwasawa main conjectures for varieties over finite fields, Ph.D. thesis, Universität

Leipzig, 2008, available at http://www.dart-europe.eu/full.php?id=162794.Wit13. M. Witte, Unit L-functions for étale sheaves of modules over noncommutative rings, preprint, arXiv:1309.2493v1,

2013.

http://www.dart-europe.eu/full.php?id=162794

On Z-zeta function

Zdzisław Wojtkowiak

Abstract We present in this note a definition of zeta function of the field Q which incorporates all p-adicL-functions of Kubota-Leopoldt for all p and also so called Soulé classes of the field Q. This zeta functionis a measure, which we construct using the action of the absolute Galois group GQ on fundamental groups.

Key words: p-adic zeta function, Galois group, fundamental group, measure

1 Introduction

In this note we propose a definition of zeta function for Q which generalizes or rather incorporates all p-adic L-series of Kubota-Leopoldt for Q. It also incorporates so called Soulé classes - which we call `-adic

polylogarithms evaluated at→10 - which are elements of H1

et(Q;Z`(n)).This is very important because the Soulé class in H1

et(Q;Z`(n)) is an analog of the real number ζ (n).Both determine an extension of Z(0) by Z(n) in different realizations.

We shall define this new Z-zeta function as a measure more precisely as a cocycle on GQ with values inmeasures. This point of view on p-adic zeta functions is indicated in [dSh, pages 15-16]. This measure ap-

pears naturally when studying the representation of the absolute Galois group GQ on πet1 (P1

Q \0,1,∞,→01).

During the meeting IWASAWA 2012 in Heidelberg Prof. John Coates mentioned the need of zeta func-tions defined on Z and with values in Z, if the author of the present note understood correctly his remark.The measure we are studying, has perhaps some properties required by him. Therefore the author decidedto write this note.

Another remark, which somebody made during his talk was that in the p-adic case one must work veryhard to prove congruence relations and then construct a measure. In this approach one gets immediatelymeasures which however depend on elements of Galois group.

Some time ago the author has planned to write together with Hiroaki Nakamura a paper on adelic poly-logarithms, which should generalize our work [NW02]. Though we never even started such a paper, thisproject served also as a motivation to write the present note.

The action of GK on πet1 (EK \0,

→0 ), where EK is an elliptic curve over K, leads also to measures (see

[Nak95] and [Nak13]). We do not know if one gets p-adic non-Archimedean zeta functions of EK in thiscase. However if the elliptic curve has a complex multiplication then it seems clear that the obtained measureleads to p-adic non-Archimedean zeta functions of the elliptic curves with a complex multiplication. Thiswas our impression when we looked at various papers on the subject, for example [Yag] or [dSh].

Zdzisław WojtkowiakUniversité de Nice-Sophia Antipolis, Département de Mathématiques, Laboratoire Jean Alexandre Dieudonné, U.R.A. auC.N.R.S., No 168, Parc Valrose - B.P.N 71, 06108 Nice Cedex 2, Francee-mail: [email protected]

331

[email protected]

332 Zdzisław Wojtkowiak

On the other side in [Woj12], studying the action of GQ on torsors of paths on P1Q \ 0,1,∞ we get

measures on (Zp)r, which should lead to p-adic non-Archimedean multi-zeta functions. We do not have

generalizations of these measures to Z⊗Q-valued measures on (Z)r.

2 Formulation of main results

Let us defineZ := lim←−Z/NZ ,

where the structure maps are the projections

Z/MNZ→ Z/NZ .

We define the Iwasawa algebraZ[[Z]] := lim←−Z/NZ[Z/MZ] ,

where the structure maps are

Z/NN1Z[Z/MM1Z]→ Z/NZ[Z/MZ] .

We define an action of the group Z× on Z[[Z]]. We define this action on finite levels. If c ∈ Z× and∑

N−1i=0 ai[i] ∈ Z[Z/NZ] then

c(N−1

∑i=0

ai[i])=

N−1

∑i=0

cai[c−1i] , (1)

where [a] denotes the class of a modulo N.Let χ : GQ → Z× be the cyclotomic character. Composing χ with the action of Z× on Z[[Z]] defined

above, we get an action of GQ on Z[[Z]]. Let

Z1(GQ, Z[[Z]])

be the set of cocycles on GQ with values in the GQ-module Z[[Z]].Using the action of GQ on the tower of coverings P1

Q \ (0,∞∪µN) of P1Q \0,1,∞ we get an element

belonging to Z1(GQ, Z[[Z]]) and which we denote by

ζQ .

In fact this element already appeared in [NW02] and before in [IS87]. In [NW02] it is called adelic Kummer-Heisenberg values.

For any rational prime p there is a projection

prp : Z[[Z]]→ Zp[[Zp]]

compatible with the action of GQ on Z[[Z]] and Zp[[Zp]]. Composing ζQ with the projection prp we get anelement which we denote

ζp ∈ Z1(GQ,Zp[[Zp]]).

In [Woj12] where we are dealing only with one fixed prime, the element ζp is denoted K1(→10). The element

ζp allows to recover p-adic L-functions of Kubota-Leopoldt. We recall definitions and results from [Woj12].

Let ω : Z×p → µp−1 ⊂ Z×p be the Teichmüller character. For x ∈ Z×p we set

[x] := xω(x)−1 .

We denote by χp : GQ→ Z×p the cyclotomic character restricted to Gal(Q(µp∞)/Q).

On Z-zeta function 333

Definition 1. Let p be a rational prime. Let 0≤ β < p−1. We define the function

Zβp : Zp×GQ→ Zp

by the formula

Zβp (1− s,σ) :=

∫Z×p

[x]sx−1ω(x)β dζp(σ)(x) .

Theorem 1. (see [Woj12, Corollary 8.3]). Let p be a rational prime. Let β be even and 0≤ β < p−1. Letσ ∈ GQ be such that (χp(σ))p−1 6= 1. Then

Zβp (1− s,σ) =

12([χp(σ)]sω(χ(σ))β −1

)Lp(1− s,ωβ ) ,

where Lp(1− s,ωβ ) is the Kubota-Leopoldt p-adic L-function.

We recall that p-adic L-functions were first defined in [KL]. The definition we used in [Woj12] is that of[Lan].

Now we shall look at functions Zβp for β odd.

Definition 2. Let s ∈ Zp. Let σ ∈ GQ and let x ∈ Zp. We define a GQ-module

Zp〈s〉

defining the action of GQ on Zp byσ(x) := [χp(σ)]sx .

Let 0≤ β < p−1. We define a GQ-module (ω(χp)

β)

defining the action of GQ on Zp byσ(x) := (ω(χp(σ)))β x .

We denote byZp〈s〉⊗Zp

(ω(χp)

β)

the Zp-module Zp equipped with the action of GQ given by

σ(x) := [χp(σ)]s(ω(χp(σ)))β x .

Observe that if k is an integer then

Zp(k) = Zp〈k〉⊗Zp

(ω(χp)

k) .In Theorem 1 we have seen that for β even the element ζQ gives non-Archimedean p-adic L-series. Thenext result shows that for β odd we get Soulé classes for the field Q (`-adic Galois polylogarithms evaluated

at→10).

Theorem 2. Let p be a rational prime and let β be an integer such that 0≤ β < p−1.

i) For a fixed s ∈ Zp, the function

GQ 3 σ 7→ Zβp (1− s,σ) ∈ Zp〈s〉⊗Zp

(ω(χp)

β)

on GQ with values in a GQ-module Zp〈s〉⊗Zp

(ω(χp)

β)

is a cocycle;ii) Let β be odd and let k be a positive integer such that k ≡ β modulo p−1. Then

Zβp (1− k,σ) = (1− pk−1)(k−1)!lk(

→10)(σ) = sk(σ) ,


where lk(→10) is the p-adic Galois polylogarithm (see [Woj05]) and sk is the Soulé class (see [Sou]).

(The point ii) of the theorem is of course well known, see [Iha86, Corollary of Theorem 10] and [Del89].The proof of the point i) of the theorem is indicated in Section 3.)

3 Measures

In this section we define the element ζQ. Let

VN := P1Q \ (0,∞∪µN)

and letf MNN : VMN →VN

be given by f MNN (z) = zM . We get a projective system of finite coverings of

V1 = P1Q \0,1,∞ .

Applying the functor πet1 we get a projective system of pro-finite groups

πet1 (VN ,

→01)N∈N . (1)

We fix an embedding Q⊂ C. Hence we have a comparison homomorphism

π1(VN(C),→01)→ π

et1 (VN ,

→01)

and therefore the natural topological generators of πet1 (VN ,

→01). We denote by

xN

(loop around 0) and byyi,N

(loop around ξ iN) for 0≤ i < N, the standard free generators of πet

1 (VN ,→01). We denote by

Γ2π

et1 (VN ,

→01)

the commutator subgroup of the group πet1 (VN ,

→01).

We have( f MN

N )∗(xMN) = (xN)M (2)

and( f MN

N )∗(yi,MN)≡ yi′,N mod Γ2π

et1 (VN ,

→01) (3)

for 0≤ i < MN and where i′ ≡ i modulo N and 0≤ i′ < N.Let pN be the canonical path on VN from

→01 to 1

N

→10, the interval [0,1]. For any σ ∈ GQ, the elements

fpN (σ) := p−1N ·σ(pN) ∈ π

et1 (VN ,

→01)

form a coherent family of the projective system (1).Here and later our convention of composing a path α from y to z with a path β from x to y will be that

α ·β is defined as a path from x to z.For N ∈ N and 0≤ i < N we define elements

αNi (σ) ∈ Z


by the congruence

fpN (σ)≡N−1

∏i=0

(yi,N)αN

i (σ) mod Γ2π

et1 (VN ,

→01) . (4)

It follows immediately from the congruences (3) that the functions

Z/NZ 3 i 7→ αNi (σ) ∈ Z (5)

form a distribution on Z with values in Z, hence they form a measure. This measure we denote by

ζQ(σ) .

The element ζQ(σ) belongs to Z[[Z]]. Hence we get a function

ζQ : GQ→ Z[[Z]] .

Definition 3. The functionζQ : GQ→ Z[[Z]]

we call Z-zeta function of the field Q.

We shall study properties of the function ζQ. First however we state a result about the action of GQ on

πet1 (VN ,

→01). Next we shall calculate explicitly the coefficients αN

i (σ).

We recall that Q⊂ C. We set

ξm := e2π√−1

m

for any m ∈ N.Below the suffix iχ(σ) at yiχ(σ),N means the unique integer 0 ≤ r < N such that iχ(σ) ≡ r modulo N.

The same remark applies to other situations when suffixes are in 0,1, . . . ,N−1.

Proposition 1. Let σ ∈ GQ. We haveσ(xN) = (xN)

χ(σ)

andσ(yi,N)≡ (yiχ(σ),N)

χ(σ) mod Γ2π

et1 (VN ,

→01)

for 0≤ i < N.

Proof. Let z be a local parameter at 0 corresponding to→01. We apply σ · yi,N · σ−1 to the germ of

(1− ξ−iχ(σ)N z)

1m at

→01. After applying σ−1 we get (1− ξ

−iN z)

1m . Then we apply yi,N and we get ξm(1−

ξ−iN z)

1m . Hence finally after acting by σ we get ξ

χ(σ)m (1− ξ

−iχ(σ)N z)

1m . The effect is the same as applying

(yiχ(σ),N)χ(σ) to (1−ξ

−iχ(σ)N z)

1m .

Proposition 2. The coefficient

αNi (σ) =

(α

N,mi (σ)

)m∈N ∈ lim←−

m(Z/mZ) = Z

is given by the formula

σ((1−ξ

−iχ(σ−1)N )

1m)

(1−ξ−iN )

1m

= ξα

N,mi (σ)

m

for 0 < i < N and by the formula

σ(N−1m ) = ξ

αN,m0 (σ)

m N−1m

for i = 0.


Proof. We act on the germ of (1− ξ−iN z)

1m by the path p−1

N ·σ(pN). Notice that the local parameter at 1

corresponding to 1N

→10 is s = N(1− z). We get that

p−1N ·σ(pN) : (1−ξ

−iN z)

1m →

σ((1−ξ

−iχ(σ−1)N )

1m)

(1−ξ−iN )

1m

(1−ξ−iN z)

1m .

On the other side it follows from the congruences (4) that we get ξα

N,mi (σ)

m (1− ξ−iN z)

1m . Hence we get the

formula for 0 < i < N.To calculate the coefficient αN

0 (σ) we act on the germ of (1− z)1m . Applying pN ·σ−1 to (1− z)

1m we get

N−1m s

1m . Next applying p−1

N ·σ we get σ(N−1m )N

1m (1− z)

1m .

We indicate the cocycle and the continuity properties. Let τ and σ belong to GQ. The it follows from theequality

fpN (τσ) = fpN (τ) · τ(fpN (σ))

(see for example the proof of [Woj04, Proposition 1.0.7]) that

αNi (τσ) = α

Ni (τ)+χ(τ)αN

iχ(τ)−1(σ) . (6)

The direct consequences of the equality (6) are the point i) of Theorem 2 and the following result.

Proposition 3.i) The functionζQ : GQ→ Z[[Z]]

is a cocycle.ii) Let k be a positive integer. We have ∫

Zp

xk−1ω(x)β dζp(τσ) =

∫Zp

xk−1ω(x)β dζp(τ)+χp(τ)

kω(χp(τ))

β

∫Zp

xk−1ω(x)β dζp(σ) .

Observe that in the point ii) of the proposition we recover the cocycle property of p-adic Galois polylog-

arithms lk(→10) (Soulé classes for the field Q), see for example [Woj05, Corollary 11.0.12.].

The proof of the continuity of the function Zβp (s,σ) is also straightforward, so we formulate only the

result.

Proposition 4. The function of two variables

Zβp : Zp×GQ→ Zp

is continuous.

Theorem 1 is the main reason that we call the cocycle ζQ, the Z-zeta function of the field Q. The paper[Woj12] is still not published. Hence we give a sketch of a proof of Theorem 1.Proof of Theorem 1.

Proof. It follows from [NW02, Proposition 3] that

lk(→10)(σ) =

1(k−1)!

∫Zp

xk−1dζp(σ)(x) . (7)

In [Woj09, Proposition 3.1] we have shown that

l2k(→10)(σ) =− B2k

2(2k)!(χp(σ)2k−1) , (8)


where B2k is the 2kth Bernoulli number. (The result is already stated in [Iha90] without proof.) It is anelementary property of the measure ζp(σ) that∫

Zp

xk−1dζp(σ)(x) =1

1− pk−1

∫Z×p

xk−1dζp(σ)(x) . (9)

It follows immediately from the equalities (7), (8) and (9) that

2(χp(σ)2k−1)

∫Z×p

x2k−1dζp(σ)(x) =−(1− p2k−1)B2k

2k.

Theorem 1 follows from the last equality.

4 Relations with Iwasawa theory

We recall that we have defined the action of Z×p ( which we identify with the group Gal(Q(µp∞)/Q) viathe cyclotomic character χp) on Zp[[Zp]] via the formula (1). This action of Z×p is the consequence of theequality

fpN (τσ) = fpN (τ) · τ(fpN (σ)) .

Let M∞ be the maximal abelian pro-p extension of Q(µp∞), which is unramified outside p. In the Iwa-sawa theory the action of Z×p = Gal(Q(µp∞)/Q) on Gal(M∞/Q(µp∞)) is given via inner automorphisms. Ifσ ∈ Gal(M∞/Q(µp∞)), τ ∈ Z×p and τ is a lifting of τ to Gal(M∞/Q) then

τ(σ) := τσ τ−1

(see [CS06, page 5]).

It follows from [Woj04] that for any τ, σ ∈ GQ we have

fpN (τστ−1) = fpN (τ) · τ(fpN (σ)) ·

((τστ

−1)((fpN (τ)−1))

Hence comparing coefficients modulo Γ 2πet1 (VN ,

→01) we get

αNi (τστ

−1) = αNi (τ)−χ(σ)αN

iχ(σ)−1(τ)+χ(τ)αNiχ(τ)−1(σ) .

Now we assume that N = pn and σ ∈ GQ(µp∞ ). Then we get

αpn

i (τστ−1) = χp(τ)α

pn

iχp(τ)−1(σ) .

Hence for σ ∈ GQ(µp∞ ), the Iwasawa action of Z×p on measures ζp(σ) is given by the formula (1).

Let n≥ m. We setKn,m :=Q(µpn)

((1−ξ

ipn)

1pm | 0 < i < pn)

andK∞ :=

⋃n≥m≥1

Kn,m .

It follows from Proposition 2 that the cocycle ζp : GQ→ Zp[[Zp]] factors through the quotient group

Gal(K∞/Q)

of GQ.Observe that K∞ is an abelian pro-p extension of Q(µp∞) which is unramified outside p. Moreover K∞

is Galois over Q. Hence we have a surjective morphism of Zp[[Z×p ]]-modules


Gal(M∞/Q(µp∞)

)→ Gal(K∞/Q(µp∞)) .

Proposition 5.i) The cocycle ζp : GQ→ Zp[[Zp]] vanishes on GK∞, hence it induces

ζp : Gal(K∞/Q)→ Zp[[Zp]] .

ii) The restriction of ζp to Gal(K∞/Q(µp∞)) is a morphism of Zp[[Z×p ]]-modules.iii) The cocycle ζp : Gal(K∞/Q)→ Zp[[Zp]] is injective.

Proof. The points i) and iii) follow from Proposition 2. The point ii) was shown in the discussion beforethe proposition when we observe that the Iwasawa action on measures ζp(σ) for σ ∈ GQ(µp∞ ) is given bythe formula (1).

Let p > 2. The action of −1 ∈ Z×p decomposes any Z×p -module M on the direct sum

M = M+ ⊕M− ,

where −1 acts on M+ as the identity and on M− as the multiplication by −1.

Proposition 6. Let p > 2. The image of Gal(K∞/Q(µp∞)

)in Zp[[Zp]] by ζp is contained in Zp[[Zp]]

−.

Proof. Let σ ∈GQ(µp∞ ). Observe that the involution induced by−1 maps the element ∑pn−1i=0 α

pn

i (σ)[i] into

−∑pn−1i=0 α

pn

−i (σ)[i]. It follows from Proposition 2 that

∑pn−1i=0 α

pn

i (σ)[i]−∑pn−1i=0 α

pn

−i (σ)[i] = 0. Hence ζp(σ) ∈ Zp[[Zp]]−.

Corollary 1. The plus part of Gal(K∞/Q(µp∞)

)is 0.

We denote by (−1)∗ the morphism induced by −1 ∈ Z×p . We denote by δ0 the Dirac distribution on Zpconcentrated at 0.

We recall the definition of the Bernoulli measure E1,c on Zp (see [Lan]). Let c ∈ Z×p . The Bernoullimeasure

E1,c =(E(n)

1,c : Z/pnZ→Qp)

n∈N

on Zp is defined by

E(n)1,c (i) =

ipn − c

〈c−1i〉pn +

c−12

for 0≤ i < pn, where 0≤ 〈c−1i〉< pn and 〈c−1i〉 ≡ c−1i modulo pn.

Proposition 7. Let p > 2 and let σ ∈ GQ. We have

ζp(σ)+(−1)∗ζp(σ) = E1,χp(σ)+1−χp(σ)

2δ0 .

Proof. It follows from [Woj12, Lemma 4.1] (see also [NW09, proof of Proposition 5.13]) that αpn

i (σ)−α

pn

−i (σ)=E(n)1,χp(σ)

(i) for 0< i< pn. On the other side E(n)1,χp(σ)

(0)= χp(σ)−12 for any n. Hence the proposition

follows.


We were hoping that studying the representation of GQ on the étale fundamental group πet1 (P1

Q \

0,1,∞,→01)pro−p we can recover the Main Conjecture (see [CS06], Theorem 1.4.3). It seems however

that it is not the case . In fact in [IS87] one can find more relations with the Main Conjecture that in thepresent paper.

On the other side the Main Conjecture should appear, manifest somewhere, when studying representa-tions of GQ on π1.

Acknowledgements These research were started in January 2011 during our visit in Max-Planck-Institut für Mathematik inBonn. We would like to thank very much MPI for support. We would like also thank to Professor C. Greither for invitationon the conference Iwasawa 2008 in Kloster Irsee. We acknowledge the financial help of the Laboratoire de Dieudonné, whichallows us to participate in the meeting Iwasawa 2012 in Heidelberg.

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