A Survey of the Hodge-Arakelov Theory of Elliptic Curves II

8/13/2019 A Survey of the Hodge-Arakelov Theory of Elliptic Curves II

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A SURVEY OF THE HODGE-ARAKELOV THEORY OF

ELLIPTIC CURVES II

SHINICHI MOCHIZUKI

Abstract. The purpose of the present manuscript is to continue the surveyof the Hodge-Arakelov theory of elliptic curves (cf. [7, 8, 9, 10, 11]) that wasbegun in [12]. This theory is a sort of Hodge theory of elliptic curves anal-ogous to the classical complex and p-adic Hodge theories, but which exists

in the global arithmetic framework of Arakelov theory. In particular, in thepresent manuscript, we focus on the aspects of the theory (cf. [9, 10, 11])developed subsequent to those discussed in [12], but prior to the conferenceAlgebraic Geometry 2000 held in Nagano, Japan, in July 2000. These devel-opments center around the natural connection that exists on the pair consistingof the universal extension of an elliptic curve, equipped with an ample linebundle. This connection gives rise to a natural object which we call thecrys-talline theta object which exhibits many interesting and unexpected prop-erties. These properties allow one, in particular, to understand at a rigorousmathematical level the (hitherto purely philosophical) relationship betweenthe classical Kodaira-Spencer morphism and the Galois-theoretic arithmeticKodaira-Spencer mor phism of Hodge-Arakelov theory. They also provide amethod (under certain conditions) for eliminating the Gaussian poles,whichare the main obstruction to applying Hodge-Arakelov theory to diophantinegeometry. Finally, these techniques allow one to give a new proof of the main

result of [7] usingcharacteristic p methods. It is the hope of the author to sur-vey more recent developments (i.e., developments that occurred subsequentto Algebraic Geometry 2000) concerning the relationship between Hodge-Arakelov theory and anabelian geometry (cf. [16]) in a sequel to the presentmanuscript.

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2 S. MOCHIZUKI

Contents

1. General Introduction 2

2. The Crystalline Theta Object 5

2.1. The Complex Analogue.2.2. The Case of Ordinaryp-adic Elliptic Curves.2.3. Integral Structures and Connections.2.4. Comparison Isomorphism at Infinity.2.5. The Associated Kodaira-Spencer Morphism.

3. Lagrangian Galois Actions 17

4. Ho dge-Arakelov Theory in Positive Characteristic 23

References 23

1. General Introduction

We begin our general introduction to the topics presented in the present manuscriptby reviewing the fundamental argument from algebraic geometrywhose arithmeticanalogueis the central goal of the Hodge-Arakelov theory of elliptic curves.

LetSbe asmooth, proper, geometrically connected algebraic curveover analge-braically closed field of characteristic zero k. Let

E S

be a family of one-dimensional semi-abelian varietieswhose generic fiber is proper.Thus, (except for a finite number of exceptions) the fibers ofE S are elliptic

curves. Let us write

S

for the finite set of points over which the fiber ofE Sfails to be an elliptic curve,and

Edef= E/S|0E

for the restriction of the sheaf of relative differentials to the zero section ofE S.Then the heightof the family E S is defined to be:

htEdef= degS(

2E )

(i.e., the degree onSof the line bundle in parentheses). The height is a measure ofthearithmetic complexityof the familyE S. For instance, the family isisotrivial(i.e., becomes trivial upon applying some finite flat base extension T S) if andonly if htE= 0.

In some sense, the most important property of the height in this context is thefact that (in the nonisotrivial case) it is universally boundedby invariants dependingonly on the pair (S, ). This bound Szpiros conjecture for function fields is as follows:


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SURVEY OF HODGE-ARAKELOV THEORY II 3

htE 2gS 2 + ||(1)

(where gS is the genus of S, and || is the cardinality of ). The proof, in thepresent geometric context, is the following simple argument: Write Mell for thecompactified moduli stack of elliptic curves overk , and Mell for the divisor atinfinity of this stack. Thus, E Sdefines a classifying morphism

: S Mell

whose (logarithmic)derivative

d: 2E=Mell/k() S()

is nonzero (so long as we assume that the family E S is nonisotrivial). Thus,since d is a generically nonzero morphism between line bundles on the curve S,

the degree of its domain (i.e., htE) is the degree of its range (i.e., 2gS 2 + ||),so we obtain the desired inequality.

Note that in the above argument, the most essential ingredient is the Kodaira-Spencer morphism, i.e., the derivative d. Until recently, no analogue of sucha derivative existed in the arithmetic case (i.e., of elliptic curves over numberfields). On the other hand:

The Hodge-Arakelov theory of elliptic curves gives rise to a nat-ural analogue of the Kodaira-Spencer morphism in the arithmeticcontext of an elliptic curve over a number field.

A survey of the basic theory of this arithmetic Kodaira-Spencer morphism, togetherwith a detailed explanation of the sense in which it may regarded as being analogousto the classical geometric Kodaira-Spencer morphism, may be found in [12].

At a more technical level, in some sense the most fundamental resultof Hodge-Arakelov theory is the following: Let E be an elliptic curve over a field K ofcharacteristic zero. Let d be a positive integer, and E(K) a torsion point oforder not dividing d. Write

Ldef= OE(d [])

for the line bundle onEcorresponding to the divisor of multiplicity d with supportat the point . Write

E E

for the universal extension of the elliptic curve, i.e., the moduli space of pairs(M, M) consisting of a degree zero line bundle M on E, together with a con-nectionM. Thus,E is an affine torsor on Eunder the module Eof invariantdifferentials on E. In particular, since E is (Zariski locally over E) the spectrumof a polynomial algebra in one variable with coefficients in the sheaf of functionson E, it makes sense to speak of the relative degree over E which we refer toin this paper as the torsorial degree of a function on E. Note that (since we arein characteristic zero) the subscheme E[d] E of d-torsion points of E maps


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4 S. MOCHIZUKI

isomorphically to the subscheme E[d] E of d-torsion points of E. Then in itssimplest form, the main theorem of [7] states the following:

Theorem 1.1. (Simple Version of the Hodge-Arakelov Comparison Isomorphism)LetEbe an elliptic curve over a fieldK of characteristic zero. WriteE E forits universal extension. Letd be a positive integer, and E(K) a torsion point

whose order does not divided. WriteLdef= OE(d []). Then the natural map

(E, L)


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Underlying all of these new developments (especially (i), (ii)) is the theory ofthe crystalline theta object, to be discussed in 2. This object is a locally freesheaf equipped with a connection and a Hodge filtration, hence is reminiscent ofthe MF-objects of [2], 2. On the other hand, many of its properties, suchas Griffiths semi-transversality and the vanishing of the (higher) p-curvatures,are somewhat different from (and indeed, somewhat surprising from the point ofview of the theory of) MF-objects. Nevertheless, these properties are of crucialimportance in the arguments that underly developments (i), (ii).

2. The Crystalline Theta Object

2.1. The Complex Analogue. We begin the discussion of this by motivatingour construction in the abstract algebraic case by first examining the complexanalogue of the abstract algebraic theory.

LetEbe anelliptic curve overC (the field of complex numbers). In this discus-

sion of the complex analogue, we shall regardEas acomplex manifold(rather thanan algebraic variety). Let us write OE (respectively,OER) for the sheaf of complexanalytic (respectively, real analytic) complex-valued functions on E. In both thecomplex and real analytic categories, we have exponential exact sequences:

0 2i Z OEexp

OE 0

0 2i Z OERexp

OER 0

Since (as is well-known from analysis)H1(E, OER) =H2(E, OER) =H

2(E, OE) =0, taking cohomology thus gives rise to the following exact sequences:

0 H1(E, 2i Z) H1(E, OE) H1(E, OE)

deg H2(E, 2i Z) = Z 0

0 H1(E, OER) deg H2(E, 2i Z) = Z 0

In other words, (as is well-known) the isomorphism class of a holomorphic linebundleon E is not determined just by its degree (which is a topological invariant),but also has continuous holomorphic moduli(given by H1(E, OE)= 0), while theisomorphism class of a real analytic line bundle is completely determined by itsdegree. Thus, in particular, the complex analytic pair (E, L) (i.e., a polarizedelliptic curve) has nontrivial moduli, and in fact, even if the moduli ofEare held

fixed,L itself has nontrivial moduli. (Here, by nontrivial moduli, we mean thatthere exist continuous families of such objects which are not locally isomorphic to

the trivial family.) On the other hand, (if we write LRdef= LOE OER , then)the real

analytic pair (ER, LR) has trivial moduli, i.e., continuous families of such objectsare always locally isomorphic to the trivial family. Put another way,

The real analytic pair(ER, LR)is atopological invariant of thepolarized elliptic curve(E, L).


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6 S. MOCHIZUKI

Note that once one admits that ER itself is a topological invariant ofE(a factwhich may be seen immediately by thinking ofER as H1(E, S1), where S1 C

is the unit circle), (one checks easily that) the fact that LR is also a topologicalinvariant follows essentially from the fact that H1(E, OER) = 0. Note that if onethinks of the universal extension E as the algebraic analogue of ER, then theanalogue of this fact in the algebraic context is given precisely by the (easily verified)fact:

H1(E, OE) = 0

(i.e., the first cohomology module of the algebraic coherent sheaf OE). In 2.3below, we would like to exploit this fact to show that the pair (E, L|E) hasa natural structure ofcrystal (valued in the category of polarized varieties, orvarieties equipped with an ample line bundle), and that by placing an appropriateintegral structureon E one may show that this crystal exists naturally not onlyin characteristic 0, but also in mixed characteristic. Thus, in summary, the analogythat we wish to assert here is the following:

topologicalinvariance of (ER, LR) (E, L|E) is a crystal

In fact, it is useful here to recall that in some sense:

The essential spirit of the Hodge-Arakelov Theory of Ellip-tic Curves may be summarized as being the Hodge theory of thepairs(ER, LR) (at archimedean primes), (E, L|

E) (with appropri-

ate integral structures to be discussed in2.3 below at non-archimedean primes), as opposed to the usual Hodge theory of anelliptic curve which may be thought of as the Hodge theory ofER

(at archimedean primes) orE

(at non-archimedean primes).In this connection, we note that the Hodge-Arakelov theory of an elliptic curve atan archimedean prime is discussed/reviewed in detail in [7], Chapter VII, 4.

2.2. The Case of Ordinary p-adic Elliptic Curves. Central to the theory ofthe mixed characteristic analogue of the ideas presented in 2.1 is the theory ofthe etale integral structure on the universal extension of an elliptic curve. In thepresent , we would like to review the definition of the etale integral structure anddiscuss its structure in the case of an ordinary p-adic elliptic curve.

Let us work over a formal neighborhood of the point at infinity on the modulistack of elliptic curves i.e., say, the spectrum of a ring of power series of the form

S def= Spec(O[[q]])

where O is a Dedekind domain of mixed characteristic and q an indeterminate.WriteS for the completion ofS with respect to the q-adic topology, and E Sfor the tautological degenerating elliptic curve (more precisely: one-dimensionalsemi-abelian scheme), with q-parameter equal to q. Then we have natural iso-morphisms

E|S = Gm; E= OS d log(U); E|S = Gm A1


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8 S. MOCHIZUKI

Then since EFn

S is a family of elliptic curves, and the classifying morphismassociated to E S is etale, it follows that EF

n

S defines a morphism:

nS :S S

One checks easily that Sdef= 1S is a lifting of the Frobenius morphism in char-

acteristic p, and that nS is the result of iterating S a total of n times (as thenotation suggests). The morphism

V :EF E

given by dividing EF by the image in EF of E[p] will be referred to as the theVerschiebung morphismassociated to E. For any n 1, the morphism

Vn :EFn

E

given by dividing EFn

by the image in EFn

ofE[pn] is easily seen to be equal (as

the notation suggests) to the n-th iterate (i.e., up to various appropriate basechanges by iterates of S) ofV. Note that the kernel ofVn may be identified withE[pn]et. In particular, it follows that Vn is etale of degreepn.

Thus, we obtain a tower

. . . EFn

EFn1

. . . EF E

of etale isogenies of degree p. Let us denote the p-adic completion of the inverselimit of this system of isogenies by EF

. Thus, in particular, we have a naturalmorphism

EF

E

In [9], 2.2, a certain canonical section

[pn] :EFn

Z/pnZ E Z/pnZ

is constructed of the universal extension ofEoverEFn

. This section is the ordinaryanalogue of the section near infinity (cf. the discussion at the beginning of thepresent 2.2) given by sendingT 0. Lettingn , we thus obtain a morphism

[p] : EF

(E)

(where denotes p-adic completion). Finally, by computing near infinity, oneshows that this morphism in fact factors uniquely through the etale integral struc-ture, i.e., determines a morphism:

et :EF (Eet)

Theorem 2.2. (Explicit Description of theEtale Integral Structure of an OrdinaryElliptic Curve) The natural morphism

et :EF (Eet)

from thep-adic completion of the Verschiebung tower ofEto thep-adic comple-tion of the universal extension ofEequipped with the etale integral structure is anisomorphism.


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10 S. MOCHIZUKI

where we denote by T : E E the morphism translation by , and weobserve that

(T L) OE([0E] []) L

(by the elementary theory of line bundles on elliptic curves), so (T L)definesa connection on L over T U, i.e., a section ofE

Eover T U, as desired. Thus,we obtain an action ofE on E which is compatible(via the projections E E,E E) with the usual translation action ofE on itself. Moreover, it is an easyexercise to show that the action of E on E defines a structure of E-torsor onE.

WhenS is Z-flat, andL is the degree one line bundle defined by atorsion point E(S), then there is also a natural analogue of the etale integral structureonE (cf. 2.2) for E. For simplicity, we restrict ourselves here to the case where = OE([0E]). Then (since (Mell)Z is connected and regular of dimension two) theetale integral structure on E is uniquely determined over arbitrary S once it is

determined near infinity, i.e., in the case S= Spec(Z[[q]]). Now, as we saw at thebeginning of2.2, thestandard integral structure onE is given (near infinity) by:

r0

OGm Tr

Relative to this notation, the standard integral structure onE is given by:

r0

OGm (T1

2)r

Thus, just as the etale integral structure onE is given by:

r0 OGm T

rit should not surprise the reader that theetale integral structure onE is given by:

r0

OGm

T 12

r

Here, we note that in the case of a more general torsion point E(O[[q1/N]])(where O is the ring of integers of some number field), the 12 is replaced by arational number of the form 12 , where Qis any representative of the unique

class in Q/Zsuch that if we think ofE(O[[q1/N]]) as Gm(O[[q1/N]][q1])/qZ via

the Schottky uniformization of the Tate curve E then is equal to the pointdetermined by q for some root of unity .

Just as in the case ofE, the etale integral structure on E extends over all of(Mell)Z. At primes other than 2, this follows immediately from the case ofE

. Onthe other hand, at the prime 2, this extension result is much more difficult to prove(cf. [9], 8.2, 8.3) and requires various complicated 2-adic computations involvinghigher p-curvatures (where p = 2). At any rate, once this extension is made, oneobtains an S-scheme (which is not of finite type over S)

Eet


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equipped with a natural structure ofEet-torsor.Next, let us assume that S is smooth of finite type over Z. In this situation,

it is proven in [9], Lemma 5.1, that the well-known connection on E determines

natural connections on Eet andEet. Moreover, by using the fact (cf. [9], Theorem

4.3) that

f(OEet

) = OS; R1f(OE

et) = 0

(where, by abuse of notation, we denote all structure morphisms to Sby f) onederives (cf. [9], Theorem 5.2) the following fundamental(i.e., relative to the theoryof [9])result(cf. the discussion of2.1):

Theorem 2.4. Let Eet(S) be a horizontal point of Eet. Then restriction to

defines a natural bijection between connections (overZ) on the pair (Eet, L|Eet)and connections on the line bundleL| onS.

Remark 2.5. Typically, in the situations that we are interested in, there will bea natural choice of connection on L|, so we shall not discuss this technical issuefurther in this survey. Thus, (once one chooses a connection on L|) there is anatural choice of connection on(Eet, L|Eet). In particular, we thus obtain a naturalcrystal valued in the category of schemes equipped with an ample line bundleon thecrystalline site of PD-thickenings ofS over Z.

Before proceeding, we observe that the crucial fact

f(OEet

) = OS; R1f(OE

et) = 0

which is immediate in characteristic 0 holds in mixed characteristic only forthe etale integral structure, not for the standard integral structure on E. This isthe technical reason for the appearance of the etale integral structure in Theorem

2.4. It is an interesting coincidence that this integral structure just happens to bethesame integral structureas the integral structure that was applied in [7] to makethe comparison isomorphism (cf. Theorem 1.1) valid in mixed characteristic.

Once one has a connection on the pair (Eet, L|Eet), the next natural step is toconsider the resulting connectionVon the locally free (albeit of infinite rank!)sheaf

V def= f(L|E

et)

onS. Note that by considering thetorsorial degreeof sections ofL overEet (i.e.,relative degree over E), one obtains a natural Hodge filtration

Fr(V) V

whose subquotients are given by:

(Fr+1/Fr)(V) = 1

r! rE OS f(L)

(wheref(L) is the push-forward of the original line bundle L on E). The triple

(V, Fr(V), V)


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12 S. MOCHIZUKI

is referred to as the crystalline theta object (cf. [9], Theorem 8.1) and is the mainobject of study in [9].

2.4. Comparison Isomorphism at Infinity. In this , we would like to takea closer look at the crystalline theta objectintroduced in 2.3 in a neighborhoodof infinity, i.e., over the base S = Spec(Z[[q]]). In this case, if we think of the

tautological elliptic curve over USdef= S[q1], i.e., the Tate curve, as the quotient

Gm/qZ

and write U for the standard multiplicative coordinate on Gm, then sections of

V def= f(L|Eet) may be thought of as linear combinations of the theta function

def=kZ

(1)k q12k2 1

2k Uk

12

(where U 1

2 is a certain natural trivialization of L over Gm, and we note thatthe exponent ofqis always integral) and its derivatives(with respect to U).

In fact, in the present context, it is natural to consider certain specific derivatives(i.e., congruence canonical Schottky-Weierstrass zeta functions cf. [9], 6) ofthe theta function, as follows. For integersr 0, let

rdef= r

2

1

2

(i.e., the greatest integer the number in brackets);

Fr(Z)def= {0 r, 1 r, . . . , r 1 r} Z

Thus, Fr+1(Z) Fr(Z) is obtained from Fr(Z) by appending one more integer

k[r] directly to the left/right ofFr

(Z) (where left/right depends only on theparity ofr ). In particular, the map Z0 Zgiven by

r k[r]

is a bijection. Also, let us write:

(k)def=

1

2k2

1

2k

Then a topological (with respect to theq-adic topology)OS-basis ofVis given byCG0 ,

CG1 , . . . ,

CGr , . . . V, where we define:

CGr = kZ k+ r

r (1)k q

12k2 1

2k Uk

12

(cf. [7], Chapter V, Theorem 4.8).Next, let us recall that the integral structure on Eet was defined by:

r0

OGm

T 12

r

hence that sending T 0 defines aq-adic section


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Gm : (Gm)

S

Eet|

S

(at least after one inverts 2). Then pulling back viaGm and applying the trivial-izationofL|Gm given by U

12 yields a natural evaluation morphism:

: V L|(Gm)S O(Gm)Swhich is, in fact, integral overZ (cf. [9], Theorem 6.1). Moreover, the connectionV is projectively compatible that is, compatible up to a possible error termconsisting of a scalar-valued (logarithmic) differential on S (cf. [9], Theorem 6.1,for more details) with the natural connection on O(Gm)S (regarded as a quasi-coherent O

S

-module) arising from the fact that (Gm)

S

arises by pulling back toSthe Z-schemeGm (i.e., put another way, the unique connection for which integral

powers ofU arehorizontal).Next, let us observe (cf. [7], Chapter V, Theorem 4.8) that the sections CGrintroduced above have the following congruence property:

(CGr ) 0 (mod q(k[r]))

Thus, it is natural to consider the sections

CGr def= q(k[r]) CGr q Vwhich define anew integral structureon V, which we denote by

VGP

where the GP stands for Gaussian poles, i.e., the poles arising from theq(k[r]) cf. [7], Chapter VI, Theorem 4.1 and the Remarks following that

theorem. In particular, theCGr form atopologicalOS-basisfor VGP, and factorsthroughVGP to form a morphism:

GP : VGP O(Gm)SNow we have the following Schottky-theoretic analogue (i.e., d-torsion points arereplaced by Gm) of the original Hodge-Arakelov comparison isomorphism (cf.,Theorem 1.1):

Theorem 2.6. (Schottky-Theoretic Hodge-Arakelov Comparison Isomorphism)Theevaluation map introduced above is an isomorphism:

GP : VGP O(Gm)S

which isprojectively horizontalwith respect to the natural (logarithmic) connectionV (cf. Theorem 2.4) on the left and the trivial connection (i.e., for which theintegral powers ofUare horizontal) on the right.


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14 S. MOCHIZUKI

Proof. This is essentially a combination of [9], Theorems 6.1, 6.2. The main ideais that, since the integral powers of U form a topological O

S

-basis of O(Gm)

S

, it

suffices to prove the result moduloq, in which case it essentially follows from the factthat the binomial coefficient functions form a Z-basis of the space of integer-valuedpolynomial functions on Z.

A number of interesting corollaries may be read off of Theorem 2.6, as follows.The first such corollary is thecomputation of the monodromyof (V, V) at the pointat infinity, i.e., V(q) S (cf. [9], Corollary 6.3). More important from the pointof view of the further development of the theory is the vanishing of thep-curvature(cf. [3], 5,6, for a discussion of the notion of p-curvature) cf. [9], Corollary6.4.

In fact, in the present context, in addition to the p-curvature, the higher p-curvaturesof the pair (V, V) may also be defined (cf. [9], 7.1). Then one has thefollowing result (cf. [9], Corollary 7.6):

Corollary 2.7. All of the higherp-curvatures of the crystalline theta object vanish(over an arbitraryZ-smooth baseS).

Proof. Note that since these higher p-curvatures all form sections oflocally freesheaves (at least in, say, the universal case S = (Mell)Z) it suffices to check thatthey are zero in a neighborhood of infinity, i.e., for the present S = Spec(Z[[q]]).Thus, we may apply the isomorphism of Theorem 2.6, so Corollary 2.7 follows fromthe fact that O(Gm)S is (topologically) generated by itshorizontal sections.

In more down-to-earth terms, Corollary 2.7 may be interpreted as follows. Itfollows from the general theory of integrable connections that in any PD formalneighborhood (where, PD stands for puissances divisees, i.e., divided powers)

of a point ofS, sections ofVmay bePD formally integratedto formhorizontal sec-tions of(V, V) in the given PD formal neighborhood. In general, such horizontalsections are not defined on a formal neighborhoodof the given point ofS, i.e., (if,for instance, t is a local coordinate on a relatively one-dimensional smooth S overZ, then) such horizontal sections are only defined once one introduces the dividedpowers:

tn

n!(wheren 0 is an integer). To state then thatall the higherp-curvatures vanishmeans that such horizontal sections exist as formal power series in t with integralcoefficients.

The above discussion motivates the following point of view. Near infinity, thehorizontal sections of the above discussion are simply the integral powers of U.Thus, the classical theta expansion

def=kZ

(1)k q12k2 1

2k Uk

12

may be thought of (in the present context) as the expansion of a generator ofF1(V) in terms of local horizontal sections of (V, V). In particular, if one takes


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16 S. MOCHIZUKI

E def= R1fDR,OE

for the first relative de Rham cohomology module of E S, and F1

(E) E(respectively, E) for its Hodge filtration (respectively, Gauss-Manin connection),thenEdefines a morphism

F1(E) E OS S/Z

which may be composed with the projection E E/F1(E) F1(E) OS

2E to

obtain a morphism:

E :F1(E) F1(E) OS

2E OS S/Z

Since F1(E) =E is a line bundle, Emay thus be regarded as a morphism:

2E S/Z

Then we have the following result (cf. [9], Theorem 8.1):

Theorem 2.10. (The Kodaira-Spencer Morphism of the Crystalline Theta Ob-ject) The Kodaira-Spencer morphism

V :2E

1

2 S/Z

associated to the crystalline theta object (V, Fr(V), V) is equal to 12 times the

classical Kodaira-Spencer morphism

E :2E S/Z

associated to the relative first de Rham cohomology module of the familyE S.

Proof.It suffices to prove this result in the universal case, i.e., when S= (Mell)Z.Moreover, the asserted equality clearly holds over all of (Mell)Z if and only if itholds in a neighborhood of infinity. Thus, we may assume that S = Spec(Z[[q]]).Now the main idea is to consider the theta expansion:

def=kZ

(1)k q12k2 1

2k Uk (U

12 )

By Theorem 2.6, the connection Vamounts (in the context of such expansions)to the (logarithmic) partial derivative q /q. On the other hand, generators ofFr(V) forr >1 are obtained by taking various derivatives of with respect to U.Since applyingq/qto thek-th term of the expansion amounts to multiplying the

k-th term by 12

k2 12k, it follows that (if we ignore the trivialization (U 1

2 ),then) applying q /q to the k-th term of the expansion gives the same result as

applying 12(U /U)2 12 (U /U) to this term. That is to say, we have:q /q

=

12

(U /U)2 1

2(U /U)

On the other hand, near infinity, the classical Kodaira-Spencer morphism amountsto the identification of q /q with (U /U)2. This completes the proof ofthe asserted equality.


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Remark 2.11. Thevanishingof thep-curvatures (cf. Corollary 2.7) in the presenceof a Kodaira-Spencer morphism which is an isomorphism (cf. Theorem 2.10) issomewhat surprising from the point of view of the classical theory of MF-objectsarising from families of varieties, in which vanishing of thep-curvature is related tovanishing of the Kodaira-Spencer morphism (cf. [4]).

3. Lagrangian Galois Actions

In this , we apply the theory of2 to

(i) show how to eliminate the Gaussian poles from (a certain version of) thearithmetic Kodaira-Spencer morphism under certain conditions (cf. Corol-lary 3.5);

(ii) show that the reduction in positive characteristic of (a certain version of)the arithmetic Kodaira-Spencer morphism coincideswith the classical ge-ometric Kodaira-Spencer morphism (cf. Corollary 3.6).

The ideas surveyed in this are discussed in more detail in [10], 2 (in the case ofodd p); [11], 3 (in the case ofp= 2). The key idea here is that the theory of thecrystalline theta objectallows one to study the (globally defined)discretearithmeticKodaira-Spencer morphism from a (local p-adic)continuouspoint of view.

Let p be an odd prime. Let K def

= Qp(p), where p is a primitive p-th root ofunity. Write OK (respectively, m; k) for the ring of integers of K (respectively,maximal ideal ofOK; residue field ofOK);

S def= Spec(OK[[q]])

and

E S

for the tautological degenerating elliptic curve (more precisely: one-dimensionalsemi-abelian scheme) with q-parameter equal to qover S. Also, let us write:

Z def= Spec(OK[[q

1p ]])

Thus, over Q, ZQ SQ is finite, etale, and Galois, with Galois group

G= Fp(1)

(where the (1) denotes a Tate twist). As is well-known, over USdef= S\V(q), the

p-torsion pointsofE Sfit into a natural exact sequence:

0 p|US E[p]|US Fp|US 0which splits over UZ

def= USSZ. In the following discussion, we consider the

particular splitting

H|UZ E[p]|UZ

(i.e., subgroup scheme H|UZ that projects isomorphically onto Fp|UZ ) of this exact

sequence defined by the p-th root ofqgiven by q1p .


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18 S. MOCHIZUKI

Next, let us writeC Sfor the uniquesemi-stable modelcompactifyingE S.Thus, C is regular, and the complement of its unique node is equal to E. On the

other hand,CZ

def

= CSZ is no longer regular. Let us denote by

CZ Z

the unique regular semi-stable modelover Z compactifying E|UZ . Thus, the com-plement

EZ Z

of the nodes ofCZhas a natural group scheme structure whose special fiber (i.e.,

fiber over V(q1p )) is a disjoint union ofp copies ofGm.

Next, let us observe thatH|UZ determines (by taking the closure in EZ) aclosed

subgroup scheme

H E

Z

ofEZ. We may then form the quotient:

EZ EZ EH

def= EZ/H

Thus,EH Z is a one-dimensional semi-abelian schemewith q-parameter equal

to q1p . Let us write

LHdef= OEH ([0EH ])

and LZdef= LH|E

Z; LZ

def= LZ|EZ . Moreover, it may be shown (cf. [10], 2.1, for

more details) that, if we denote by EZ(Z) the unique section of order 2, thenalthough a priori,

L def= LZ|

appears only to be defined overZ, in fact, itadmits a naturalG-action(compatiblewith theG-action on OZ), as if it arose as the pull-back to Zof a line bundle onS.

On the other hand, note (cf. the exact sequence discussed above) that p maybe regarded as a closed subgroup scheme:

(p)Z EH

In addition, one has a natural closed subgroup scheme

{1} EH

so we shall write (p)Z EH for the translateof (p)Zby the element 1.

If we now substitute the pair (EH, LH) into the theory of 2.3, 2.4, we obtainthe corresponding crystalline theta object

(VH, Fr(VH), VH )

over Z by considering sections ofLHover the Hodge torsor equipped with its etaleintegral structure EH,et associated to this pair. Moreover, it follows immediatelyfrom the definition of the integral structure on EH,et that the subgroup schemes(p)Z, {1} EHlift naturallyto closed subschemes


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(p)Z, {1} EH,et

hence that we obtain a natural lifting

(p)Z EH,et

of (p)Z EH. On the other hand, by the theory of theta groups(cf., e.g., [7],Chapter IV, 1), we have a natural theta trivialization

LH|(p)Z L OZ O(p)Z

of the restriction ofLH to (p)Z.In particular, if we restrictsections ofLH overE

H,et to (p)Zand then apply

the theta trivialization, we obtain a morphism

VH : VH L OZ O(p)Z

whose restriction

HDR : HDR L OZ O(p)Z

to HDRdef= Fp(VH) VH is essentially (i.e., up to taking H-invariants) the

restriction morphism appearing in the original Hodge-Arakelov Comparison Iso-morphism (cf. Theorem 1.1). That is to say, it follows from the main theorem of[7] that:

HDR |UZ is an isomorphism.

(In order to make it an isomorphism over Z, one must introduce the Gaussianpoleson HDR.) In particular, since the Galois groupG acts naturally on the rangeofHDR , we get a natural action ofG on the domain of HDR , at least over UZ.

Definition 3.1. This action ofG on HDR|UZ will be referred to as the LagrangianGalois actionon HDR.

Remark 3.2. Note, in particular, that, unlike the Galois actions considered in [7],Chapter IX, 3; [10], 1 (cf. [12], 1.4.1, for a survey of this theory), theLagrangianGalois action depends on the choice of the subgroup H. In fact, however, thechoice of different splittingsHdoes not affect the present theory very much. What

is, however, essential here, is the existence of the natural multiplicative subgroupscheme:

p|US E[p]|US

Indeed, this is the essential element of the present discussion in a neighborhoodof infinity that does not exist in the number field case. We will say more on thislater (cf. Remark 3.7 below).


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Corollary 3.5. (Elimination of Gaussian Poles) The Lagrangian Galois action onHDR|UZ is, in fact, defined onHDR (i.e., without Gaussian poles).

Finally, we would like to apply Theorem 3.4 to relate the Lagrangian arithmeticKodaira-Spencer morphism to the classical Kodaira-Spencer morphism. To do this,we must consider the Lagrangian Galois action modulo m2. First, we observe thatit follows immediately from Theorems 2.8, 3.4 that for any G, the LagrangianGalois action of onHDR maps

F1(HDR) F3(HDR) OK (OK/m

2)

in such a way that modulo m, F1(HDR) maps into F1(HDR). In particular, if we

compose this map with the projection F3 F3/F2 (cf. the discussion of 2.5),then we obtain an OZ-linear morphism

F1

(HDR) OK k (F3

/F2

)(HDR) OK (m/m2

) (2EH OZ F

1(HDR)) OK (m/m2)

i.e., if we let G vary, we obtain a homomorphism

G: G = Fp(1)

HomOZ (F1(HDR) OK k, (

2EH

OZ F1(HDR)) OK (m/m

2))

= 2EH OK (m/m2)

On the other hand, if we forget about families of elliptic curvesand form the naturalexact sequence of differentials associated to the triple Zlog Slog Spec(OK),we obtain an extension ofG-modules:

0 OK/Zp OK OZ= (p1 m OZ) OK (OK/m

p1) Zlog/Zp Zlog/Slog = Zlog/OK Fp 0

(where all differentials involving Slog,Zlog are understood to beq-adically continu-ous). By considering the extent to which sections of Zlog/OKFplift toG-invariantsections of Zlog/Zp , we thus obtain a homomorphism

G Fp(Zlog/OK Fp) (p1 m OZ) OK (OK/m

p1)

whose composite with the morphism induced by the projection m m/m2 is easilyverified to be an isomorphism

: G FpOZk Zlog/OK OK m/m

2

(cf. the theory of [1], I., 4)relating the Galois groupG to the (logarithmic) tangent

bundleZlog/OKdef= HomOZ (Zlog/OK , OZ) modulo m.

Thus, if we combine with G, we obtain a morphism:

G : Zlog/OK OK k 2EH

OK k


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22 S. MOCHIZUKI

On the other hand, if we compute G by means of Theorems 2.10, 3.4, it followsthat:

Corollary 3.6. (Relation to the Classical Kodaira-Spencer Morphism) The mor-phism

G : Zlog/OK OK k 2EH

OK k

constructed by considering the Lagrangian Galois action modulo m2 and applyingthe natural isomorphism

: G FpOZk Zlog/OK OK m/m

2

between the Galois group G and the (logarithmic) tangent bundle modulo mis equalto 12 the classical Kodaira-Spencer morphism associated to the familyEH Z.

Remark 3.7. From the point of view of applications to diophantine geometry (cf.[12], 1.5.1), one would like to develop an analogue of the theoryof the present overnumber fields i.e., as opposed to over formal neighborhoods of infinity (cf.

the base S def= Spec(OK[[q]])), as in the present discussion. Indeed, as discussed in[12], 1.5.1, the main essential obstruction to applying Hodge-Arakelov theory todiophantine geometry is the elimination of the Gaussian poles i.e., the numberfield analogue of Corollary 3.5. As discussed above (cf. Remark 3.2), the mainingredient that one needs in order to realize such a theory over number fields is ananalogue over number fields of the multiplicative subgroup scheme:

p|US E[p]|US i.e., a global multiplicative subspace of the Tate module (of an elliptic curveover a number field). A first step towards realizing such a global multiplicativesubspace is taken in [10], 3, where a global multiplicative subspace is constructedover the base:

(Mell)Q

It is then proposed inloc. cit. that perhaps byrestrictingthis subspace over (Mell)Qto a Q-valued point, one may achieve the goal of constructing a global multiplicativesubspace for elliptic curves over Q. Unfortunately, this operation of restriction isnot so straightforward, since it involves numerous intricacies related to the issue ofkeeping track of basepoints(of the various fundamental groups involved). Moreover,this issue of keeping track of basepoints appears to be related to Grothendiecksanabelian geometry. At the time of writing, the author is optimistic that by applyingthe anabelian theory of [15]; [16]; and [18], Chapter XII, 2, these technical problemsmay be resolved and hence that a natural global multiplicative subspace for ellipticcurves over number fields may be constructed. It is the hope of the author to expose

these ideas (cf. [17]) in a future sequel to the present manuscript.


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4. Hodge-Arakelov Theory in Positive Characteristic

In this , we explain how a certain extension of the theory of2.2 for ordinary

elliptic curves to supersingular elliptic curves in positive characteristic can be usedto give an alternate proof of Theorem 1.1. More details on the theory discussedhere may be found in [11], 1, 2.

Let

E S

be a family of elliptic curvesover an Fp-schemeSsuch that the associated classifyingmorphism

S (Mell)Fp

is etale. In the following discussion, we will write S : S S for the Frobeniusmorphismon S, and denote the result of base-changing objects over S via S by

means of a superscript F. Thus, the morphism

[p] :E E

(multiplication by p) factors as a composite of morphisms:

E E EF

V E

where E is the relative Frobenius morphismofE S, and V is the Verschiebungmorphism. As is well-known, the kernels of E and V are Cartier dual to oneanother:

E[E] EF [V]

Next, let us observe that the pull-back

VE EF

of theuniversal extensionE E viaVadmits a canonical section:

: EF VE

Indeed, if we can show that the E-torsor VE EF istrivial locally onS, then

may be defined as the unique section that takes the origin 0EF ofEF to the origin

ofVE (which is determined by the origin 0E ofE). Thus, it suffices to show

that the E-torsor in question is trivial locally on S. Since R1f(E) (where, by

abuse of notation, we denote all structure morphisms to S by f) is a line bundleon S, it suffices to prove this local triviality in the case that E S is ordinary.

But then, V

induces an isomorphism E

EF , so it follows that the morphism(again induced by V)

OS=R1f(E|E) R

1f(EF |EF ) = OS

(where the isomorphisms on the two ends are the trace maps of the residues andduality theory ofE, EF) is given by multiplication by p = deg(V). Since we areworking in characteristicp, this completes the proof that theE-torsor V

E EF

is trivial (locally on S).


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24 S. MOCHIZUKI

Next, let us consider structure sheaves. Let us regard OE as a quasi-coherentsheaf ofOE-algebras(via the projectionE

E). Similarly, V :EF Eallows usto regard OEF as a coherent sheaf ofOE-algebras. Thus, the morphism : E

F VE induces a morphism of quasi-coherentOE-modules:

: OE OEF

Let us write O


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det(OEF ) = det(O

EF

[V]) = det(O

E[E ]) =

p1

j=0 j [E] = 1

2p(p 1) [

E]

(since E[E] E is just an infinitesimal neighborhood of the origin 0E). Thiscompletes the proof.

Remark 4.2. A version of Theorem ?? where < p is replaced by < pn (wheren 1 is an integer) is given in [11], Theorem 1.1.

Next, we consider line bundles. Let

E(S)

be a torsion point of order prime to p. WriteF def= E() EF(S) and set:

LFdef= OEF ([

F ]); Mdef= E(L

F)

Thus, LF (respectively, M) is a line bundle of degree1 (respectively, degreep) onEF (respectively, E). Since the order of isprime to p, it follows thatF does notintersect0EF , hence that we have an isomorphism

f(LF)

LF |0

EF

given by restricting sections over EF to 0EF . Now we are ready to apply themethod of the above discussion to prove the following positive characteristic versionof Theorem 1.1:

Theorem 4.3. (Positive Characteristic Version of the Hodge-Arakelov Compari-son Isomorphism) The natural restriction morphisms

EF :f(LF|VE) L

F|EF[V]

andE :f(M|[p]E) M|E[p]

induced by are isomorphisms.

Proof. First, we observe that by the theory of theta groups(applied to descentvia the Frobenius morphism E :E E

F) the bijectivity of EF is equivalenttothat of E. We refer to [11], 2, for more details.

Next, we observe that this discussion extends to the case ofdegenerating ellipticcurves (cf. [11], 2, for more details). Thus, we may work over the proper stack(Mell)Fp , so (just as in the proof of Theorem ??) it suffices to show that EF

is generically an isomorphismand that the determinant bundlesassociated to itsdomain and range define the same element of Pic((Mell)Fp)Q.

The fact that EF (or, equivalently, E) is an isomorphism over an open densesubstack of (Mell)Fp follows from the theory of the comparison isomorphism nearinfinity developed in [7], Chapter V (cf. [11], 2, for more details).

On the other hand, the isomorphism

f(LF)

LF |0

EF


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26 S. MOCHIZUKI

implies that (just as in the proof of Theorem ??) the determinants of the domainand range of EF are both equal to:

p1j=0

j [E] = 1

2p(p 1) [E]

This completes the proof.

Remark 4.4. Thus, although in the above proof of Theorem ??, we use the (rel-atively easy) portion of the theory of [7] concerning the comparison isomorphismnear infinity, the intricate degree computations of [7], Chapter VI, are notused inthe proof of Theorem ??. Also, we observe that although Theorem?? is a positivecharacteristic result, reduction modulo p implies Theorem 1.1, at least in the cased = p. Moreover, since the truth of the characteristic zero result Theorem 1.1 forarbitraryd is equivalent to the equality of the degrees of two specific line bundles,both of which are polynomials ind (cf. the complicated degree computations of [7],Chapter VI), the fact that these two polynomials are equal whenever d is a primenumber implies that they are equal for alld. Thus, in summary, the above techniqueyields a simple proof of Theorem 1.1 for arbitraryd. We refer to [11], 2, for moredetails.

Remark 4.5. One way to interpret the preceding Remark ?? is the following:

The characteristicp methods (involving the Frobenius and Ver-schiebung morphisms) of the above discussion yield a new proof ofthe various combinatorial identities inherent in the computation ofdegrees in [7], Chapter VI, proof of Theorem 3.1.

This situation is rather reminiscent of the situation of [14], Chapter V cf.,

especially, the second Remark following Corollary 1.3. Namely, in that case, aswell, characteristic p methods (involving Frobenius and Verschiebung) give rise tovarious nontrivial combinatorial identities. It would be interesting if this sort ofphenomenon could be understood more clearly at a conceptual level.

Remark 4.6. One interesting feature of the above proof is the crucial use of theFrobenius morphismE :E E

F. Put another way, this amounts to the use of thesubgroup schemeE[E] E(i.e., the kernel of E), which, of course, does not existin characteristic zero. Note that this subgroup scheme is essentially the same as themultiplicative subspacethat played an essential role in the theory surveyed in 3.That is to say, it is interesting to note that just as in the context of3, thecrucialarithmetic object that one wants over a number field is a global multiplicativesubspace, in the above proof, the crucial arithmetic object that makes the proof

work (inpositive!characteristic) is the global multiplicative subspaceE[E] E(which is defined over all of (Mell)Fp).

Remark 4.7. Another interesting and key point in the above proof is the fact that,unlike the case in characteristic zero (where the structure sheaf of a finite flat groupscheme on a proper curve always has degree zero):

In positive characteristic, the structure sheaf of a finite flatgroup scheme on a proper curve can have nonzero degree.


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In fact, it is precisely because of this phenomenon that in order to make the com-parison isomorphism hold in characteristic zero over the proper object (Mell)Q, itis necessary to introduce the Gaussian poles(cf., e.g., [7], Introduction, 1).

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Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502,

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A Survey of the Hodge-Arakelov Theory of Elliptic Curves II

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