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Journal of Number Theory 130 (2010) 1068–1097

Contents lists available at ScienceDirect

Journal of Number Theory

www.elsevier.com/locate/jnt

Iwasawa theory of totally real fields for certain non-commutative p-extensions Takashi Hara ∗,1 Graduate School of Mathematical Sciences, The University of Tokyo, 8-1 Komaba 3-chome, Meguro-ku, Tokyo 153-8914, Japan

a r t i c l e i n f o a b s t r a c t

Article history: Received 5 August 2009 Available online 23 October 2009 Communicated by P. Schneider

MSC: 11R23 11R80 19B28

Keywords: Non-commutative Iwasawa theory Whitehead groups

In this paper, we will prove the non-commutative Iwasawa main conjecture—formulated by John Coates, Takako Fukaya, Kazuya Kato, Ramdorai Sujatha and Otmar Venjakob (2005)—for certain specific non-commutative p-adic Lie extensions of totally real fields by using theory on integral logarithms introduced by Robert Oliver and Laurence R. Taylor, theory on Hilbert modular forms introduced by Pierre Deligne and Kenneth A. Ribet, and so on. Our results give certain generalization of the recent work of Kazuya Kato on the proof of the main conjecture for Galois extensions of Heisenberg type.

© 2009 Elsevier Inc. All rights reserved.

Contents

0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070 2. The main theorem and Burns’ technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073 3. The additive theta map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075 4. Translation into the multiplicative theta map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1080 5. Localized theta map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087 6. Congruences among abelian p-adic zeta pseudomeasures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089 7. Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097

* Fax: +81 (03) 5465 7011. E-mail address: thara@ms.u-tokyo.ac.jp.

1 The author is supported by JSPS Core-to-Core Program 18005, “New Developments of Arithmetic Geometry, Motive, Galois Theory, and Their Practical Applications.”

0022-314X/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jnt.2009.08.008

http://www.ScienceDirect.com/ http://www.elsevier.com/locate/jnt mailto:thara@ms.u-tokyo.ac.jp http://dx.doi.org/10.1016/j.jnt.2009.08.008

T. Hara / Journal of Number Theory 130 (2010) 1068–1097 1069

0. Introduction

The Iwasawa main conjecture, which predicts the mysterious relation between “arithmetic” char- acteristic elements and “analytic” p-adic zeta functions, has been proven under many situations for abelian extensions of number fields. However, it took many years even to formulate the main conjec- ture for non-commutative extensions. In 2005, John Coates et al. [5] formulated the main conjecture for elliptic curves without complex multiplication by using algebraic K -theory. Then Kazuya Kato [13] has constructed the p-adic zeta functions and proven the main conjecture for certain specific p-adic Lie extensions called “Heisenberg-type extensions” of totally real number fields. Mahesh Kakde [11] has recently generalized Kato’s method and proven the main conjecture for another type of exten- sion. On the other hand, Jürgen Ritter and Alfred Weiss had also formulated the main conjecture for 1-dimensional p-adic Lie extensions called “equivariant Iwasawa main conjecture” [19] in a little dif- ferent way from the formulation of John Coates et al., and proven it for certain special cases [20–22]. In this paper, we will prove the Iwasawa main conjecture—the formulation of John Coates et al.— for another type of non-commutative p-extensions of totally real number fields by generalizing the method of Kazuya Kato [13].

Let p be a prime number, F a totally real number field and F ∞ a totally real p-adic Lie extension of F containing the cyclotomic Zp-extension of F . Assume that only finitely many primes of F ramify in F ∞ . For simplicity, also assume that Iwasawa’s μ = 0 conjecture is valid (see condition (�) in Section 1.2 for more general condition). The aim of this paper is to prove the following theorem.

Theorem 0.1 (= Theorem 2.1). Assume the following two conditions:

(1) the Galois group G of F ∞/F is isomorphic to ( 1 Fp Fp Fp

0 1 Fp Fp 0 0 1 Fp 0 0 0 1

) × Γ where Γ is a commutative p-adic Lie

group isomorphic to Zp ; (2) the prime number p is not equal to either 2 or 3.

Then the p-adic zeta function ξF ∞/F for F ∞/F exists and the Iwasawa main conjecture is true for F ∞/F .

We will review the formulation of the non-commutative Iwasawa main conjecture in Section 1.2. Now let us summarize how to prove the theorem. The basic strategy to construct the p-adic zeta functions and prove the main conjecture for non-commutative extensions has been proposed by David Burns, and Kazuya Kato applied it in his results [13]. We will also use their technique to prove our main theorem (Theorem 0.1); precisely we first construct the family F consisting of pairs {(Ui, V i)}4i=0 of subgroups of G such that each V i is normal in Ui and all Artin representations of G are obtained as Z-linear combinations of representations induced by characters of the abelian groups {Ui/V i}4i=0 (see Section 3.1 for details). Let Λ(G) (resp. Λ(Ui), Λ(Ui/V i)) denote the Iwasawa algebra of G (resp. Ui , Ui/V i). Consider the composition

θS,i : K1 ( Λ(G)S

) Nri−−→ K1(Λ(Ui)S) canonical−−−−−→ K1(Λ(Ui/V i)S) = Λ(Ui/V i)×S for each i where Λ(G)S (resp. Λ(Ui)S , Λ(Ui/V i)S ) denotes the canonical Ore localization of Λ(G) (resp. Λ(Ui), Λ(Ui/V i)) introduced by John Coates et al. in [5, Section 2] (also refer to Section 1.2 of this article) and Nri denotes the norm map. Set θS = (θS,i)4i=0. Then each Λ(Ui/V i)×S contains the p-adic zeta pseudomeasure ξi for F V i /FUi constructed by Pierre Deligne, Kenneth A. Ribet and Jean-Pierre Serre [6,24] (FUi and F V i denote the maximal intermediate fields of F

∞/F fixed by Ui and V i respectively). Now suppose that there exists such an element ξ in K1(Λ(G)S ) as θS (ξ) coincides with (ξi)4i=0. Then we may check that ξ satisfies the interpolation property which characterizes the p-adic zeta function for F ∞/F (see Definition 1.5) by using Brauer induction, hence the element ξ is nothing but the p- adic zeta function for F ∞/F . Moreover ξ satisfies “the main conjecture” ∂(ξ) = −[C F ∞/F ] because the main conjecture for each abelian extension F V i /FUi holds by virtue of the deep results of Andrew Wiles [29] (see Section 2.2 for details). Therefore we have only to verify that (ξi)4i=0 is contained in the image of the map θS . However it is difficult to characterize the image of θS . On the other hand,

1070 T. Hara / Journal of Number Theory 130 (2010) 1068–1097

we may characterize the image Ψ of θ = (θi)4i=0 by using theory on integral logarithms introduced by Robert Oliver and Laurence R. Taylor [17,18] (see Section 3 and Section 4), where θi is the composition

θi : K1 ( Λ(G)

) Nri−−→ K1(Λ(Ui)) canonical−−−−−→ K1(Λ(Ui/V i)) = Λ(Ui/V i)×. By this fact and easy diagram chasing, we may conclude that (ξi)4i=0 is contained in the image of θS if it is contained in a certain subgroup ΨS of

∏ i Λ(Ui/V i)

× S (which contains the image of θS ) charac-

terized by certain norm relations and congruences (see Theorem 2.4 and Section 5 for details). Now the proof of Theorem 0.1 reduces to the verification of norm relations and congruences for

(ξi) 4 i=0. We may easily verify the norm relations by formal calculation using the interpolation proper-

ties for {ξi}4i=0. To study congruences among {ξi}4i=0, we will use the q-expansion principle for Hilbert modular forms proven by Pierre Deligne and Kenneth A. Ribet [6] (see Section 6). Kazuya Kato, Jür- gen Ritter and Alfred Weiss also used the q-expansion principle and successfully derived congruences among abelian p-adic zeta pseudomeasures which were necessary to verify their results [13,20–22]. Unfortunately it is difficult to derive all the desired congruences in our case by only using the tech- nique of Kato, Ritter and Weiss. Therefore we will use the existence of the p-adic zeta function for a certain non-commutative subextension F N/F of F ∞/F which is of Heisenberg type (note that the existence of the p-adic zeta functions for Heisenberg-type extensions has already been proven by Kazuya Kato [13]), and complete the proof of our Theorem 0.1 by using certain induction (see Sec- tion 7 for details).

Notation In this paper, p always denotes a positive prime number. We denote by N the set of natural

numbers (that is, the set of strictly positive integers), denote by Z (resp. Zp) the ring of integers (resp. p-adic integers), and denote b

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