The Galois-Theoretic Kodaira-Spencer Morphism of an Elliptic Curve

8/13/2019 The Galois-Theoretic Kodaira-Spencer Morphism of an Elliptic Curve

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The Galois-TheoreticKodaira-Spencer Morphism of an Elliptic Curve

Shinichi Mochizuki

July 2000

Contents:

0. Introduction

1. Galois Actions on the Torsion Points

2. Lagrangian Galois Actions

2.1. Definition and Construction

2.2. Relation to the Crystalline Theta Object

3. Global Multiplicative Subspaces

4. The Group Tensor Product

Section 0: Introduction

The purpose of this paper is to study in greater detail the arithmetic Kodaira-Spencer morphismof an elliptic curve introduced in [Mzk1], Chapter IX, in the gen-eral context of theHodge-Arakelov theory of elliptic curves, developed in [Mzk1-3].In particular, after correcting a minor error (cf. Corollary 1.6) in the construction ofthis arithmetic Kodaira-Spencer morphism in [Mzk1], Chapter IX, 3, we define (cf.2.1) a slightly modified Lagrangian version of this arithmetic Kodaira-Spencermorphism which has the following remarkable properties:

(1) This Lagrangian arithmetic Kodaira-Spencer morphism is freeof Gaussian poles (cf. Corollary 2.5).

(2) A certain portion of the reduction modulo pof this Lagrangianarithmetic Kodaira-Spencer morphism may be naturally iden-tified with the usual geometric Kodaira-Spencer morphism (cf.Corollary 2.7).

We recall that property (1) is of substantial interest since it is the Gaussian polesthat are the main obstruction to applying the Hodge-Arakelov theory of ellipticcurves to diophantine geometry (cf. the discussion of [Mzk1], Introduction, 5.1,

for more details). On the other hand, property (2) is of substantial interest in that

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2 SHINICHI MOCHIZUKI

it shows quite definitively that the analogy asserted in [Mzk1], Chapter IX, betweenthe arithmetic Kodaira-Spencer morphism of the Hodge-Arakelov theory of ellipticcurves and the usual geometric Kodaira-Spencer morphism of a family of ellipticcurves is not just philosophy, but rigorous mathematics! (cf. the Remark followingCorollary 2.7 for more details). In fact, both properties (1) and (2) are essentially

formal consequences of a property that we refer to as the crystalline nature ofthe Lagrangian Galois action (cf. Theorem 2.4). Interestingly, the theory of2 ofthe present paper makes essential use not only of the theory of [Mzk1], but also of[Mzk2], [Mzk3].

Unfortunately, however, this Lagrangian arithmetic Kodaira-Spencer morphism,which is based on a certain Lagrangian Galois action, can only be defined whenthere is a natural (rank one) multiplicative subspace(i.e., weight 1 subspace) ofthe Tate module of the elliptic curve in question. Since such a subspace is well-known to exist in a formal neighborhood of infinity (of the compactified modulistack of elliptic curves), we work over such a base in 2. Ultimately, however, onewould like to carry out this construction for elliptic curves over number fields. In3, 4, we discuss a certain point of view that suggests that this may be possible cf. especially, 4, Conclusion. It is the hope of the author to complete theconstruction motivated in3,4 in a future paper.

Notation and Conventions:

We will denote by (Mlog

1,0)Z the log moduli stack of log elliptic curves overZ(cf. [Mzk1], Chapter III, Definition 1.1), where the log structure is that definedby the divisor at infinity. The open substack of (M1,0)Z parametrizing (smooth)

elliptic curves will be denoted by (M1,0)Z (M1,0)Z.

Acknowledgements:The author would like to thank A. Tamagawa and T. Tsuji forstimulating discussions of the various topics presented in this manuscript.

Section 1: Galois Actions on the Torsion Points

In this , we study variousGalois actionson the space of functions on the setof torsion points of an elliptic curve. In particular, we observe that these actionsgive rise to a natural action of the algebraic fundamental groupoid of the baseof a family of elliptic curves on the scheme of torsion points over this base. Thisaction allows us to correct an errormade in [Mzk1], Chapter IX, in the definitiongiven there of the arithmetic Kodaira-Spencer morphism.

Let Slog be a fine noetherian log schemewhose underlying scheme S is con-nected and normal. Write USS for the open subscheme where the log structureis trivial. In the following discussion, we shall assume thatUS = . Next, let usassume that we are given alog elliptic curve(cf. 0, Notations and Conventions)

Clog Slog


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GALOIS-THEORETIC KODAIRA-SPENCER 3

(whose associated one-dimensional semi-abelian scheme we denote by ES) anda positive integer d 1 which is generically invertible on S (i.e., invertible on aschematically dense open subscheme ofS). Next, let us write

UT[d1

] E|US[d1]

(where US[d1]

def= USZ Z[d1]) for the kernel of the finite, etale morphism [d] :

E|US [d1] E|US[d1]of degreed2 (given by multiplication byd). Thus,UT[d

1]US[d

1] is finite etale of degree d2. Let us write

TS

for the normalizationofS in UT[d1], UT

def= T|US (so UT[d

1] = UTZ Z[d1]).

Also, let us assume that we are given a connected, normal scheme Z over S such

that if we writeUZdef= Z|US , UZ[d

1]def= UZZ Z[d

1], then UZ[d1] US[d

1] isfinite etale and Galois; UZ[d

1] US[d1] dominates every connected component

ofUT[d1]; and Z is the normalizationofS in UZ[d

1].

Next, we consider etale fundamental groups. Write

S

for the fundamental group 1(US[d1]) (for some choice of base-point, which we

omit in the notation since it is irrelevant to our discussion). Then since UZ[d1]

US[d1] is Galois, it follows that S acts naturally on UZ[d1], hence also on Z(over S). In particular, Sacts naturally on the module ofd-torsion points

M def= MorS(Z, T)

(where we observe that, as an abstractZ-module, M=(Z/dZ)2). In the followingdiscussion, we shall think of the action of SonZas an action from the right, andthe action of S onOZ, Mas an action from the left.

Next, let us us consider the OZ-algebra

F def= Func(M, OZ)

ofOZ-valued functions on the finite set M. Then observe that S acts onF inseveral different ways, e.g., via the action of SonM, via the action of SonOZ,etc.

Definition 1.1. We shall refer to the OZ-linear (respectively, semi-linear, relativeto the action of S on OZ) action of S from the right (respectively, left) on F

induced by the action of S on M(respectively, OZ) as the point-theoretic action


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(respectively,value-theoretic action) of S on F. We shall refer to theOZ-semi-linear action of Son Ffrom the left given by composing the value-theoretic actionwith the inverse of the point-theoretic action (i.e., if S, and F, then maps to the function M m ((1m))) as the diagonal actionof S onF. We shall abbreviate the term point-theoretic action (respectively, value-

theoretic action; diagonal action) by the term P-action(respectively, V-action;D-action).

Proposition 1.2. There is a natural inclusion: OT F ofOS-algebras whichinduces an isomorphism of OT onto the subalgebra of F of S-invariants withrespect to the D-action. Moreover, this inclusion induces an isomorphism

(OTOS OZ)norm F

(where the superscript norm denotes the normalization of the ring in parentheses)which is S-equivariant with respect to the tensor product of the trivial action ofS onOT and the natural action of S onOZ on the left, and the D-action ofS onFon theright.

Proof. It is a tautology (cf. the definitionM def= MorS(Z, T)) that elements ofM

defineOS-algebra homomorphismsOT OZ. Thus, if we take the direct productof these various homomorphisms, then we get an OS-algebra homomorphism

: OT F

(cf. the definition ofF i.e.,F def= Func(M, OZ)). Moreover, it is a tautology that

OTmaps into the subalgebra of S-invariants ofF relative to the D-action. SinceOT and F are both normal algebras, in order to verify the remaining assertionsof Proposition 1.2, it suffices to verify that these assertions hold over US[d

1].To simplify notation, we assume (just for the remainder of this proof) that S =US[d

1].

Thus,T S and Z S are finite etale, so, by (Galois) etale descent (withrespect to the morphism Z S), it follows that the S-invariants of F form

an OS-algebra OT , whose spectrum T

over S is finite etale over S. Moreover,T S factors (cf. the discussion of the preceding paragraph) throughT. Thus,since T and T are both finite etale of degree equal to the cardinality ofM (i.e.,d2) overS, we obtain thatT =T, as desired. The fact that the induced morphism

OTOS OZFis an isomorphism then follows from elementary properties of etale

descent. This completes the proof.

Next, we would like to use the V-action of S on F to construct some sortof group action on F which descends to an action on OT. If, for instance, theV-action of S on F were equivariantwith respect to the D-action of S on F,

then the V-action itself would define an action of S on F that descends to an


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action onOT. In fact, however, this sort of equivariance does not hold in the naivesense, but only in the following twisted sense:

First, let us define the profinite group withS-actionS as follows: We takethe underlying profinite group of Sto be S itself. The S-action on Sis defined

to be the action given by: ()def

= 1 (for S,S). Next, we endowF with the S-actiondefined by thinking of S as Sand using the V-actionofS onF (cf. Definition 1.1). Then we have the following:

Proposition 1.3. TheS-action onF is compatible with theS-actions onSandF (where we think ofSas acting onFby the D-action).

Proof. Indeed, if S, S, F, m M, then:

D{()}(m) = {()}(1 m)= (1 m)

= 1 (1 m)

=() (1 m)

=() {D()}(m)

=

()

{D()}

(m)

(where the superscript D denotes the D-action of the group element bearing the

superscript). This completes the proof.

Let us write Gal(Z/S) for the Galois group of the Galois covering UZ[d1]

US[d1]. Then by forming the quotient ofZby the action of Gal(Z/S)in the sense

of stacks, we obtain an algebraic stackSskZ, together with morphisms

ZSskZ S

where the first morphism isGalois, finite etale(with Galois group Gal(Z/S)), and

the second morphism is an isomorphism overUS[d1

]. If we let Z range over allconnected Galois, finite etale coverings ofS, the inverse limit of the Z(respectively,

SskZ) thus defines a pro-schemeS(respectively,pro-algebraic stackSsk) overS,

together with morphisms:

S Ssk S(where the first morphism is Galois, profinite etale (with Galois group S) andthe second morphism is an isomorphism overUS[d

1]). Moreover, by etale descent,the profinite group with S-action Sdefines a profinite etale group scheme

skS

over Ssk.


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Definition 1.4. We shall refer to skS as the algebraic fundamental groupoidofSlog.

Note that sinceZ SskZ isfinite (Galois) etale, the D-action of SonSpec(F)

(where Spec is to be understood as being taken with respect to the structure ofOZ-algebra on F) definesdescent datafor Spec(F) with respect toZSskZ (hence,

a fortiori, with respect toSSsk). We shall denote the resulting descended objectover Ssk by:

Tsk Ssk

Note that over US[d1], we haveTsk|US [d1] =T|US [d1] (by Proposition 1.2).

Corollary 1.5. The action of S on F descends to an action of the algebraicfundamental groupoidskS onTsk.

Proof. This follows immediately from Proposition 1.2, 1.3.

Remark. The terminology of Definition 1.4 may be justified as follows: First,let us recall the well-known analogy between algebraic fundamental groups (suchas S) and the usual topological fundamental groups of algebraic topology. Thisanalogy gives us the freedom to phrase our justification in the language oftopologicalfundamental groups. Thus, letXbe atopological manifold. Note that for each point

x X, we obtain (in a natural way) a group:

x 1(X, x)

i.e., the fundamental group with base-point x. This correspondence defines a localsystem of groups onX, which is known (in algebraic topology) as the fundamentalgroupoidofX. On the other hand, there is a well-known equivalence of categoriesbetween local systems of groups on X and groups G equipped with an actionof 1(X, x) (given by associating to such a local system its fiber at x, togetherwith the monodromy action of 1(X, x)). Moreover, it is an easy exercise to

check that, relative to this equivalence of categories, the fundamental groupoidcorresponds to the group with 1(X, x)-action defined by letting 1(X, x) acton itself via conjugation(cf. the definition given above for the S-action on S).Thus, in summary, one may regard the object skS as the algebraic analogue of thefundamental groupoid. This justifies the terminology of Definition 1.4.

Remark. One other interesting observation relative to the appearance of the al-gebraic fundamental groupoid (cf. Definition 1.4 i.e., as opposed to group) inthe correct formulation of the arithmetic Kodaira-Spencer morphism (cf. Corollary1.6 below) is the following. Recall that in the assertedanalogybetween the arith-

metic Kodaira-Spencer morphism of [Mzk1], Chapter IX, and the usual geometric


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Kodaira-Spencer morphism, the Galois group/fundamental group(oid) of the baseplays the role of the tangent bundle of the base. On the other hand, the tangentbundle of the base (typically) does not admit a canonical global trivialization, butinstead varies from point to point i.e., at a given point, it consists ofinfinites-imal motions originating from that point. Thus, it is natural that the arithmetic

analogue of the tangent bundle should be not the static fundamental group, butinstead the fundamental groupoid, which varies from point to point, and indeed,at a given point, consists ofpaths(which may be thought of as a sort of motion)originating from that point.

We are now ready to apply the above discussion to correct an errormade in[Mzk1], Chapter IX, in the definition given there of thearithmetic Kodaira-Spencermorphism:

In [Mzk1], Chapter IX, Theorem 3.3, and the discussion preced-ing it, it is falselyasserted that there is a natural action (withdenominators) of S on HDR (notation of loc. cit.) arisingfrom a natural action of S on T S (notation of the presentdiscussion).

Although this assertion does indeed hold if the S-schemeThappens to be a disjointunion of copies ofS, in general, it is false. That is to say, the correct formulationof this assertion is that the twisted object skS acts on T

sk (not that Sacts onT).

Corollary 1.6. (Correction to Error in [Mzk1], Chapter IX, 3) Thephrase natural action of S on ... HDR in [Mzk1], Chapter IX, Theorem 3.3,should read

natural action ofskS on ... HDR|Ssk

(where |skS denotes the pull-back via the morphismSsk S), and the divisor of

possible poles [

(dE)] + V(4) should read

[

(dE)] + V(d)

In particular, the resultingarithmetic Kodaira-Spencer morphism arithE :S Filt(HDR)(S) ((erroneous) notation of [Mzk1], Chapter IX, 3) should infact be thought of as a morphism

arith, skE : skS Filt(HDR)|Ssk

of sheaves on the etale site ofSsk.

Proof. It remains only to remark that the reason for the divisor V(d) (i.e., thezero locus of the regular function determined by the integer d) is that the integral

structure on the pull-back to Z(notation of the present discussion) ofOdE differs


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from that ofF(cf. the notation norm appearing in Proposition 1.2) by a factorwhich is bounded by the differentof the algebra O

dE. Moreover, the fact that thisdifferent dividesd follows from the fact that multiplication by d on an elliptic curveinduces multiplication byd on the differentials of the elliptic curve.

Remark. Thus, in summary, what is done in [Mzk1], Chapter IX, 3, is literallycorrect whenTShappens to be a disjoint sum of copies ofS(a condition whichmay be achieved by base-change). In general, however, there is a certaintwistthat must be taken into account, but which was overlooked in the discussion of[Mzk1], Chapter IX.

Section 2: Lagrangian Galois Actions

In this, we study the Galois action on the torsion points of an elliptic curve,along with the resulting arithmetic Kodaira-Spencer morphism (cf. 1)under theassumption that this Galois action preserves a rank one multiplicative submodule of the module of torsion points. In this situation, we show that the resultingLagrangian Galois action is defined without Gaussian poles, and, moreover, thata certain piece of the resulting arithmetic Kodaira-Spencer morphism coincideswith the classical Kodaira-Spencer morphism (cf. 2.2). This further strengthensthe analogy discussed in [Mzk1], Chapter IX, between the arithmetic (or Galois-theoretic) Kodaira-Spencer morphism and the classical (geometric) Kodaira-

Spencer morphism.

2.1. Definition and Construction

In this , we maintain the notation of1. Moreover, we assume that we aregiven a S-submoduleM

Mwhose underlying Z/dZ-submodule isfree of rankone. Thus, we obtain an exact sequence of S-modules

0 M MMet 0

(where Met is defined so as to make this sequence exact). Thus, restricting OZ-valued functions on M to M gives rise to a surjection

F= Func(M, OZ) F def= Func(M, OZ)

ofOZ-algebras. Observe that the V-action (cf. Definition 1.1) of S on F mani-festly preserves this surjection, so we get a natural V-action ofS onF

. Since,moreover, we are operating under the assumption that Spreserves the submoduleM M, it thus follows that the P- and D-actions (cf. Definition 1.) of S on

F also preserve this surjection, so we also get natural P- and D-actions ofS on


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F. In particular, (cf. Proposition 1.2) taking the spectrum (over S) of the S-invariants ofF with respect to the D-action gives rise to a scheme T togetherwith a morphism

T

T

which is aclosed immersion overUS[d1]. (Note that since the operation of taking

S-invariants is not necessarily right exact, it is not clear whether or not T T

is a closed immersion over S.)

Next, let us suppose that we are given a splitting

MH M

of the surjection of modules M Met

(i.e., a submodule MH

M such thatthe morphism MH Met is bijective), which is not necessarily preserved byS.Even ifMH is not preserved by S, however, since M

M is preserved by S,it follows that an element S will always carryM

H M to another splittingMH

Mof the surjectionM Met.

Next, we return to the de Rham point of view, and consider sections ofline bundles on the universal extensionofE. Also, for simplicity, we assume fromnow on that S is Z-flat, and that d is odd. Then we would like to consider theHodge-Arakelov Comparison Isomorphism (cf. [Mzk1], Introduction, Theorem A),so, in the following discussion, we will use the notation of [Mzk1], Introduction,

and [Mzk3], 9. (Here, we recall that certain minor errors in [Mzk1], Introduction,Theorem A were corrected in [Mzk3], 9, Theorem 9.2.) Thus, we assume that wehave been given an integerm that does not divided, together with a torsion point

E,S(S)

of order preciselym which defines ametrized line bundleLst, onE,S (cf. [Mzk1],Chapter V, 1). Here, we recall that S is the stack (in the finite, flat topology)obtained from S by gluing together US (away from infinity) to the profinitecovering ofS(near infinity) defined by adjoining a compatible system ofN-throots of theq-parameter (asN ranges multiplicatively over the positive integers).Over S, we have the group object

E,S S

which is equal to ESover US(away from infinity), and whose special fiberconsists of connected components indexed by Q/Z, each of which is isomorphic toa copy ofGm cf. the discussion of [Mzk1], Chapter V, 2, for more details. For

simplicity, we shall also assume that n def= 2m is invertible onS. In the following

discussion, we shall simply write L forLst,. Thus, in particular, over US:


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L|US =OE(d [])|US

Also, let us write

Ed,Z Z

for the object which is equal to EZdef= ESZ over UZ, and, near infinity, is the

pull-back to Zof the object Ed (cf. [Mzk1], Chapter IV, 4, where we take Nof loc. cit. to be d). (In words, this object Ed is the complement of the nodes ofthe unique regular semi-stable model of the Tate curve (withq-parameter q) over

the base Z[[q1d ]].) Then the object E[d],et E of [Mzk3], 9, defines an object

E

[d],et,Z

Ed,Z

(which, over (UZ)Q, may be identified with the universal extensionE EofE)

over Ed,Z. Indeed, the discussion of [Mzk3], 9, applies literally over UZ; nearinfinity, the fact that we get an object over Ed,Z follows from the fact that the

integral structure in question, i.e.,

d(T(i/2m))r

(in the notation of [Mzk3], 9)

is invariant with respect to the transformations T T+ jd , j Z.

Note that L has an associated theta group (cf. [Mzk1], Chapter IV, 1, 5, fora discussion of theta groups) GZ over Zwhich fits into an exact sequence:

1 (Gm)Z GZ dEZ1

(where dEZ (Ed,Z) Z is the finite flat group scheme ofd-torsion points). Letus suppose that the submodulesM, MH Marise from the restriction toUZ[d

1]offinite flat group schemes

GZ, HZ dEZ

over Zsuch that the resulting morphism GZZHZ dEZ is an isomorphism of

group schemes. Thus, (cf. Proposition 1.2) we have anatural inclusion

OGZ

F

of finite OZ-algebras, which is an isomorphism over UZ[d1]. In the following

discussion, we alsoassume thatGZhas been chosen so that the subalgebraOGZ F is preserved by the various actions (i.e., V-, P-, D-) ofS onF. Finally, letus assume that we are given a lifting

HZ GZ


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ofHZ (i.e., HZHZ viaGZ dEZ). (Thus, HZ is a Lagrangian subgroup (cf.

[MB], Chapitre V, Definition 2.5.1) of the theta groupGZ.) In particular, we get anatural action ofHZ=HZ onL.

In the following discussion, we will always denote (by abuse of notation)struc-

ture morphisms to S, Z, E,S byf(cf. the conventions of [Mzk1]). We would liketo consider the push-forward

VLdef= f(LE

[d],et,Z)

of the pull-back LE[d],et,Z

of the metrized line bundle L to E[d],et,Z. (Here, we

take the integral structure of this push-forward near infinity to be the uniqueGZ-stable integral structure determined by the

CGr cf. [Mzk1], Chapter V,

Theorem 4.8; the discussion of [Mzk3], 4.1, 4.2.) Thus,VLis aquasi-coherent sheafonZ. Also, often we would like to consider thefiltrationFr(VL) VL consisting

of sections whose torsorial degree is< r. (Here, by torsorial degree, we meanthe relative degree with respect to the structure of relative polynomial algebraon OE over OE(arising from the fact that E

E is an affine torsor). Since E

may be identified with E[d],et,Zover (UZ)Q, this definition also applies to sections

ofVL.) In particular, we shall write

HDRdef= Fd(VL)

for the object which appears in [Mzk1], Introduction, Theorem A (cf. also [Mzk3],Theorem 9.2). Thus,HDR is a vector bundle of rankd onZ. Finally, observe thatthe theta group GZacts naturally onVL, F

r(VL), HDR.

Next, let us observe that (by the assumption that n= 2m is invertible on S)the d-torsion subgroup scheme dEZEd,Z lifts (uniquely!) to a subgroup scheme:

dE E[d],et,Z

(Indeed, this follows from the fact that the integral structure used to define E[d],et,Zis given by

d(T(i/2m))

r

(in the notation of [Mzk3], 9), an expression which

gives integral values OS for all T = j

d

(for j Z).) Moreover, we recall from[Mzk1], Chapter IX,3, that (since d is odd) we have a theta trivialization

L|dEZ

=L|0EZ OZ OdEZ

(where 0EZ EZ(Z) is the zero section ofEZdef= ESZ Z) of the restriction of

Lto dEZEd,Z. Note, moreover, sinceL is defined over E,S(i.e., without base-changing to Z), it follows that L|0EZ is, in fact, defined over S (i.e., in otherwords, it is defined over S, except that near infinity, one may need to adjoinroots of theq-parameter). In particular, it follows that there is a natural action of

Gal(Z/S) hence ofS(via the surjectionS Gal(Z/S)) onL0EZ .


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Thus, by restricting sections ofL over E[d],et,Z to dE E[d],et,Z, and com-

posing with the theta trivialization reviewed above, we obtain a morphism:

V :VL L|0EZ OZ OdEZ

Similarly, if we introduce Gaussian poles (cf. [Mzk1], Introduction, Theorem A,(3); [Mzk3], Theorem 6.2), we get a morphism:

GPV :VGPL

L|0EZ OZ OdEZ

Then the main result of [Mzk1] may be summarized as follows:

Theorem 2.1. (Review of the Main Result of [Mzk1]) Assume thatd is

odd and thatn def= 2m is invertible on S. Then restriction of sections ofL overE[d],et,Z to thed-torsion points, composed with the canonical theta trivialization

ofL over thed-torsion points yields a morphism

V :VL L|0EZ OZ OdEZ

whose restrictionH :HDR L|0EZ OZ OdEZ

to HDRdef= Fd(VL) VL satisfies: (i) H is an isomorphism overUZ; (ii) if one

introducesGaussian poles, i.e., if one considers

GPH : HGPDR L|0EZ OZ OdEZ

thenGPH is an isomorphism overZ.

Proof. We refer to [Mzk1], Introduction, Theorem A, especially (2), (3). Notethat the zero locus of the determinant is empty because of our assumption thatn is invertible on S.

Next, let us observe that it follows from the fact that the morphism GZZHZ dEZ is an isomorphism of group schemesthat the composite

OHZdEZ

OdEZ OGZ

(where OHZdEZ

denotes the subalgebra ofOdEZ of functions which are invariant with

respect to the natural action ofHZon OdEZ ) is anisomorphism. Thus, by applyingthis isomorphism to the various morphisms obtained by taking HZ=HZ-invariants

of the various morphisms of Theorem 2.1, we obtain the following result:


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Corollary 2.2. (Lagrangian Version of the Main Result of [Mzk1])

Assume thatd isodd and thatndef= 2misinvertible on S. Then by applying the

isomorphismOHZdEZ

OdEZ OGZ to the result of takingHZ

=HZ-invariants ofthe various morphisms of Theorem 2.1, we obtain a morphism

HZV : VHZL

L|0EZ OZ OGZ

whose restrictionHZH : H

HZDR L|0EZ OZ OG

Z

to HHZDRdef= Fd(VHZ

L ) VHZ

L satisfies: (i) HZH is an isomorphism overUZ; (ii) if

one introducesGaussian poles, i.e., if one considers

GP,HZH : HGP,HZDR L|0EZ OZ OG

Z

thenGP,HZH is an isomorphism overZ.

Proof. This follows from Theorem 2.1, together with the elementary observationthat taking the HZ = HZ-invariants of an HZ = HZ-equivariant isomorphism isagain an isomorphism.

Before proceeding, we recall that VHZL

admits the following interpretation:

Since HZ = HZ acts on Ed,Z; E,S; E[d],et,Z; L, we may form the quotients of

these objects by this action. This yields objects (Ed,Z)H, (E,S)H, (E

[d],et,Z)H,(L)H(a metrized line bundle on (E,S)H). Then we have:

VHZL

=f{(L)H|(E[d],et,Z)H}

(wherefas usual denotes the structure morphism toZ) cf., e.g., [Mzk1], ChapterIV, Theorem 1.4.

Definition 2.3. The V-, P-, and D-actions of S on GZ, together with the

isomorphism GP,HZH of Corollary 2.2, and the natural action of Son L|0EZ , define

V-, P-, and D-actions of S onHGP,HZDR , which we shall refer to as theLagrangian

Galois actions onHGP,HZDR .

Remark. Thus, unlike the naive Galois actions of 1, the Lagrangian Galoisactions depend on the choice of the additional dataM, MH.

Remark. Thus,a priori, the Lagrangian Galois actions appear to require theGauss-ian poles(i.e., it appears that they are not necessarily integrallydefined onHHZDR).

In fact, however, we shall see in 2.2 below that (under certain assumptions) the


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Lagrangian Galois D-action has the remarkable property that it is defined with-out introducing the Gaussian poles. This property isnotpossessed by the naive(non-Lagrangian) Galois actions of1.

2.2. Relation to the Crystalline Theta ObjectIn this , we continue to use the notations of 2.1, except that we further

specialize them as follows. Let A be a complete discrete valuation ring of mixedcharacteristic (0, p), with perfect residue field. Write K(respectively, k) for thequotient field(respectively, residue field) ofA. In this, we suppose that S is ofthe form:

S= Spec(A[[q 1N ]])

for some positive integer N prime to d, and that the log structure on S is thatdefined by the divisor V(q

1N ) S. Also, we suppose that the given log elliptic

curve Clog Slog is the Tate curve, i.e., that it has q-parameter equal toq OS.

Thus, if we takeA sufficiently large, we may assume that Zis of the form:

Z= Spec(A[[q 1Nd ]])

and that the log structureon Zis that defined by the divisor V(q 1Nd ) Z. Thus,

we obtain a morphismZlog Slog of log schemes.

WriteEZdef= ESZ. Thus,EZZis a one-dimensional semi-abelian scheme

over Z. Note thatEZhas a unique finite flat subgroup scheme annihilated by d.This subgroup scheme is naturally isomorphic to d. We take this subgroup schemefor our

d=G

Z dEZEZ

(and note that it is easy to see that this GZsatisfies the condition(cf. 2.1) that

OGZ F be preserved by the V-, P-, and D-actions of S on F). Moreover,since (dEZ)/G

Z is naturally isomorphic to the constant group scheme (Z/dZ)Z, it

is easy to see that, over Z, there exists a finite etale group schemeHZ dEZ suchthat the natural morphism

GZZHZ dEZ

is an isomorphism of group schemes. Thus, if we writeEHZdef= (Ed,Z)H, then we

see that EHZ Z is a one-dimensional semi-abelian group scheme(i.e., its fibers

are all geometrically connected), and that the natural quotient morphism


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(EZ) Ed,Z EHZ

(over Z) has kernel equal to HZ, hence is finite etale of degreed. Finally, we notethat the q-parameter ofEHZ is a d-th root ofq. This completes our description of

the specializations of the objects of2.1 that we will use in the present.

Next, we would like torelate the present discussion to the theory of connectionsin [Mzk3]. To begin with, we recall that L|0EZ (where 0EZ EZ(Z) is the zerosection of EZ Z) is a line bundle on Z equipped with a natural S-action;moreover, this S-action is derived from the fact that L|0EZ in fact arises from aline bundle on S. Thus, there exists a trivialization

:L|0EZ=q

aNd OZ

(wherea is a nonnegative integer< N d) i.e., in the terminology of the discussionof [Mzk3], 5, a rigidification of L at 0EZ which is S-equivariant. In thefollowing discussion, wefixsuch a S-equivariant rigidification .

Observe that defines, in particular, a logarithmic connection on the linebundle L|0EZ which isstable under the action ofS. Thus, according to the theoryof [Mzk3], 5, this rigidification gives rise to (logarithmic) connections

VL

, VHZ

L

on VL, VHZL , respectively (cf. [Mzk3], Theorems 5.2, 8.1), which are stabilized

by the action of S. Here, the logarithmic connections are relative to the logstructure ofZlog, and all connections, differentials, etc., are to be understood asbeing continuous with respect to the (p, q)-adic topology onOZ.

Moreover, sinceall higherp-curvatures of these connections vanish(cf. [Mzk3],7.1, for a discussion of the general theory of higherp-curvatures; [Mzk3], Corollary7.6, for the vanishing result just quoted), we thus conclude that VL, V

HZL

define

crystals on the site

Inf(Z

log

k/A)

ofinfinitesimal thickenings over A of open sub-log schemes of Zlog k = Zlog (A/mA). That is to say, usually, crystals are defined on sites ofPD-thickenings(cf., e.g., [BO], 6, for a discussion of this theory), but here, since all of the higherp-curvatures vanish, we obtain crystals on the site of thickenings which do notnecessarily admit PD-structures. Since the definitions and proofs of basic propertiesof such crystals on Inf are entirely similar to the divided power case, we leave theunenlightening details to the reader. Thus, in summary, we may think of the pairs

(VL, VL); (V

HZ

L , VHZ

L )


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as crystals on Inf(Zlog k/A).

In the following discussion, we will not use the entire group S, but only itsm-inertia subgroup mS S, defined as follows:

(S) m

Sdef= { S | () (mod mA OZ), OZ}

Thus, in particular, one verifies easily that the image Galm(Z/S) of mSin Gal(Z/S)is a p-group. More precisely, if we write

d= dp d=p

(where dp, d=p are positive integers; dp is a power of p; and d=p is prime to p),then one verifies immediately (using the simple explicit structures ofS,Z) that the

correspondence

S (q 1Nd )/q

1Nd

defines isomorphisms:

mS Galm(Z/S)

(Z/dpZ)(1)

S Gal(Z/S) (Z/dZ)(1)

(where the vertical arrows are the natural inclusions, and the horizontal arrows aredefined by the correspondence just mentioned).

Next, let us observe that it follows immediately from the definition of mS thatevery mS defines an A-linear isomorphism

: Zlog Zlog

which is the identity on Zlog

k. It thus follows from:

(i) the fact that Zlog defines a(n) (inductive system of) thickening(s) in thecategory Inf(Zlog k/A); and

(iii) the fact that (VHZL

, VHZ

L

) forms a crystal on Inf(Zlog k/A)

that induces a -semi-linear isomorphism

:VHZL VHZL


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(where -semi-linear means semi-linear with respect to the action ofon OZ, andthe hat denotesp-adic completion). Here, the justification for the notation

is that this isomorphism is the analogue of the isomorphism obtained in differential

geometry by parallel transporting i.e., integrating sections of

VHZL

along

the path (where we think of as an element of the (algebraic) fundamentalgroup S). That is to say, we obtain a natural S-semi-linear action ofS onVHZL

. (In fact, the same holds for VHZL

replaced by VL, but since this action ismore interesting forVHZ

L , we restrict ourselves to the case ofVHZ

L in the following

discussion.)

Theorem 2.4. (Crystalline Nature of the Lagrangian Galois Action)

The action ofSonVHZL is compatible withHZV (cf. Corollary 2.2; here the hatdenotes p-adic completion) and the D-action on GZ in the following sense: For S, the following diagram commutes:

VHZL

HZ

V L|0EZ OZ OGZ D

VHZL

HZ

V L|0EZ OZ OGZ

(whereD denotes the result of applying toOGZ

via the D-action ofSonOGZ

).

Proof. This follows from the naturality of all the morphisms involved, together

with the compatibility (cf. [Mzk3], Theorem 6.1) over GZof the connectionVHZL

with the theta trivialization reviewed in2.1.

Corollary 2.5. (Absence of Gaussian Poles in the Lagrangian GaloisAction) Relative to the objects of the present discussion, the Lagrangian Galois D-

action ofS onHGP,HZDR (cf. Definition 2.3) is definedwithout Gaussian poles,

i.e., it arises from an action ofS onHHZDR .

Proof. This follows immediately from the commutative diagram of Theorem 2.4,together with Lemma 2.6 below: Indeed, since the morphism

isintegral(i.e., inparticular, defined without Gaussian poles), restricting this commutative diagram

to HHZDR VHZL

(i.e., where VHZL

denotes the VHZL

in the upper left-hand corner

of the diagram) shows that by computing the Lagrangian D-actioninsideVHZL

and

then applying Lemma 2.6 to the lower horizontal arrow of the diagram, we obtainthe asserted integrality.

Lemma 2.6. The image of the morphismHZV

(cf. Corollary 2.2) is the same

as the image of its restrictionHZH to H

HZDR VHZL .


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Proof. In a word, these two images coincide because both images are equal tothe image of the theta convolution (studied in detail in [Mzk2], cf. especially 10).Indeed, this is essentially the content of the proof of [Mzk2], Theorem 10.1. In this

proof, only the image of HZH i.e., more concretely, the span of the derivativesof the theta function of order < d is discussed, and this image is shown to

be the same as that of the theta convolution; but the argument never uses that thederivatives are of order < d i.e., the exact same argument shows that the imageof HZ

V (= the span of the derivatives of the theta function of arbitrary order)

is equal to the image of the theta convolution.

For the convenience of the reader, however, we review the argument briefly asfollows: If we write U for the standard coordinate on Gm, then the image of

HZH

(respectively, HZV

) may be identified with the span of the restrictions to d Gm

of the derivatives ( U

)j where

def= kZ

q1d ( 12 k2+(i/n)k) Uk (k)

(and(a character Z n),i(an integer) are invariants associated to the torsionpoint) oforder< d(respectively,arbitrary order). On the other hand, a basisof the functions on d is given by U

0, . . . , Ud1. If one computes the coordinates(relative to this basis) of |d , one sees that the coefficient of the smallest powerofqappearing in each coordinate is eithera root of unity ora root of unity timesa nonzero sum of two n-th roots of unity (cf. [Mzk2], Theorem 4.4, where we takedord to be 1). Since we have assumed thatn is invertible onS(cf. Theorem 2.1),

we thus obtain that this coefficient is always aunit. On the other hand, to give anelement in the span in question is to consider the restriction to d Gm of a series

kZ

q1d

( 12 k2+(i/n)k) P(k) Uk (k)

whereP() is a polynomial (with coefficients in OZ Q, but which maps Z intoOZ) of degree < d(respectively, arbitrary degree) in the case of

HZH (respectively,

HZV

). Thus, in short, the assertion of Lemma 2.6 amounts to the claim that the

span of

{P(k) q(k) Uk}k=0,... ,d1

(for some function : {0, . . . , d 1} 1Nd Z) is the same, regardless of whetherone restricts the degree ofP(() to be < d or not. But this is easily verified.

Finally, we observe that:

Theorem 2.4 allows us to relate the arithmetic Kodaira-Spencermorphism arising from the Lagrangian Galois action to the clas-

sical geometric Kodaira-Spencer morphism.


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as follows: Let

Galm(Z/S)

be asubgroup of order> 2. Writeddef

= ||for the order of . Thus, d= 1 dividesdp, and we have a natural isomorphism=(Z/dZ)(1) (cf. the discussion aboveimmediately following the definition of mS). Write

(p A) m A

for the ideal generated by elements of the form 1 , where is a d-th root ofunity. Note that acts trivially onZ (A/m) (mod m). Moreover, we have ahomomorphism

: d(A) m/m2

given by 1 (mod m2). Thus, if we think ofd(A) as (Z/dZ)(1) (whichis naturally isomorphic to ), then we see that defines a homomorphism

: m/m2

which is easily seen (by the definition of the ideal m) to induce an injection (Z/pZ)m/m2.

Next, we would like to consider a certain portion of the arithmetic Kodaira-Spencer morphism associated to the Lagrangian Galois action, which will turn outto be related to the classical Kodaira-Spencer morphism. Let . Then since acts onHHZDR via the Lagrangian Galois D-action (Definition 2.3, Corollary 2.5), we

see that defines a morphism D : HHZDR HHZDR . If we restrict this morphism to

F1(HHZDR ), and compose with the surjectionHHZDR {H

HZDR/F

2(HHZDR )} A(A/m2),

we thus obtain a morphism

F1(HHZDR) {HHZDR/F

2(HHZDR )} A(A/m2)

which (as one verifies easily by using the fact that is the identity onHHZDR A k,

HHZDR m/m2) vanishes on m F

1(HHZDR) and maps into {HHZDR/F

2(HHZDR)} Am/m

2). Thus, we obtain a morphism

:F1(HHZDR ) k {H

HZDR/F

2(HHZDR)} Am/m2

which we would like to analyze by applying Theorem 2.4. If one reduces the com-mutative diagram of Theorem 2.4 modulo m2, it follows that may be computed

by applying the (logarithmic) connectionVHZ

L

onVHZL

in the logarithmic tangent

direction


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log(q 1Nd )

()

where we observe that

(q 1Nd ) q

1Nd + () q

1Nd (mod m2)

(cf. the definition of, given above). That is to say, ApplyingVHZL

in this

logarithmic tangent direction to F1(HHZDR ) k = F1(VHZ

L ) k VL k and then

projecting viaVHZL

k VHZL

/F2(VHZL

) k defines a morphism:

F1(VHZL

) k VHZL

/F2(VHZL

) m/m2

On the other hand, by [Mzk3], Theorem 8.1 (i.e., the property which was calledGriffiths semi-transversality in loc. cit.), this morphism in fact maps into the

submodule F3(VHZL

)/F2(VHZL

) m/m2. Moreover, (F3/F2)(VHZ

L ) is naturally

isomorphic to 12 F1(VHZ

L ) 2EHZ

(where we writeEHZ for the tangent bundle to

EHZ at the origin 0EHZ ). In particular, we obtain thatalso maps into F3().

(Note that here we use that d dpd 3.) Thus, by letting vary, we obtaina homomorphism:

: HomOZ (F1(HHZDR ) k, (F3/F2)(HHZ

DR ) m/m2) =122EHZ m/m

2

arising from theLagrangian Galois action, taken modulo m2, which, when regardedas an element

Hom(,1

2 2EHZ

m/m2) = Hom(, m/m

2)

1

2 2EHZ

is equal to the result of evaluating the geometric Kodaira-Spencer morphism of thecrystalline theta object (VHZ

L ,

VHZ

L

) (cf. [Mzk3], Theorem 8.1)

: (Zlog/A)

1

2 2EHZ

on log(q

1Nd )

(Zlog/A) and multiplying the result by . We summarize this

discussion as follows:

Corollary 2.7. (Relation to the Classical Geometric Kodaira-SpencerMorphism) Let

Galm

(Z/S)


21/28


be asubgroup of order >2. This subgroup gives rise to a natural idealm A (minimal among ideals modulo which acts trivially on Zlog) and a naturalmorphism

: m/m2

(defined by considering the action of onq

1Nd

modulo m2

). Then the morphism

: HomOZ (F1(HHZDR ) k, (F

3/F2)(HHZDR ) m/m2) =

1

2 2EHZ

m/m2

obtained purely from theLagrangian Galois D-action of on HHZDR (cf. Defi-

nition 2.3, Corollary 2.5) by restricting this action to F1(HHZDR) and then reducingmodulo m2 coincides with the morphism obtained by evaluating the geometric

Kodaira-Spencer morphism of the crystalline theta object (VHZL

, VHZ

L

)

(cf. [Mzk3], Theorem 8.1)

: (Zlog/A)

1

2 2EHZ

in the logarithmic tangent direction log(q

1Nd )

(Zlog/A) and multiplying the

result by . Moreover, by [Mzk3], Theorem 8.1, this Kodaira-Spencer morphismassociated to the crystalline theta object coincides (up to a factor of 12 ) with the usualKodaira-Spencer morphism associated to the Gauss-Manin connection on the first deRham cohomology group. Thus, in summary, thearithmetic Kodaira-Spencermorphismassociated to the Lagrangian Galois D-action coincides modulo m2(andup to a factor of 12 ) with theusual Kodaira-Spencer morphism.

Remark. Note, moreover, that the correspondence between the logarithmic tangentdirection

log(q1

Nd)(Zlog/A)

and the morphismis essentially the same as the

correspondence arising from Faltings theory of almost etale extensions between thelogarithmic tangent bundle ofZlog and a certain Galois cohomology group (cf., e.g.,[Mzk1], Chapter IX,2, especially Theorem 2.6, for more details). In particular:

Corollary 2.7 shows that the analogy discussed in [Mzk1], Chap-ter IX, between the (usual)geometric Kodaira-Spencer mor-phism(cf. especially, [Mzk1], Chapter IX, Theorem 2.6) and thearithmetic Kodaira-Spencer morphism is more than just

philosophy it is rigorous mathematics!

(cf. also the discussion of [Mzk3], Theorem 6.2, in [Mzk3], 0).

Section 3: Global Multiplicative Subspaces

In this , we show how to construct a sort of global analogue of the crucialsubgroup scheme GZ dEZ of 2.2. The author believes that this construc-tion indicates the proper approach to globalizing the local theory of 2 cf. 4,

Conclusion.


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In this , let us write MZ for the moduli stack (M1,0)Z (cf. 0, Notationsand Conventions);MZ MZ for the open substack parametrizing smooth ellipticcurves. We denote base change to Q by a subscript Q. Also, let us write

MQdef

= 1(MQ)

(for some choice of base-point).

Next, let us fix a prime number p. Thus, the p-power torsion points of thetautological elliptic curve over MQ define a p-adic Tate module

F

i.e.,F is a free Zp-module of rank two equipped with a continuous MQ-action

F : MQ GL(F). Write

TQ MQ

for theprofinite etale coveringdefined by the subgroup Ker(F) MQ . SinceF issurjective(cf., e.g., [Shi], Theorem 6.23, together with the fact that the cyclotomiccharacter Gal(Q/Q) Zp

is surjective), it follows thatTQ MQ isGalois, withGalois group

Gal(TQ/MQ)=GL2(Zp)

(where the isomorphism is determined by a choice ofZp-basis forF). Write

TZ MZ

for the normalizationofMZ inTQ; TZdef= TZMZ MZ. Also, in the following dis-

cussion, we will denote the p-adic formal schemes(or stacks) defined by p-adicallycompleting various schemes (or stacks) by means of a superscript .

Now note that the natural MQ-actions onTZ and Fdefine a naturalMQ -

actionon

O

TZZpF

(which we regard as a coherent sheaf onTZ). Let us writeE for the line bun-dle on MZ defined by the cotangent bundle at the origin of the tautological logelliptic curve overMZ. Then we recall the following result, which is essentially an

immediate corollary of the p-adic Hodge theory of elliptic curves:


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Theorem 3.1. (Global Multiplicative Subspace)There is a naturalMQ -equivariant morphism

:O

TZZpF E| TZ

whose image contains (1 p) E|

TZ, where p is a primitivep-th root of unity.

Moreover, away from the supersingular points ofTZFp, this morphism is a surjec-tion, and (in a formal neighborhood of infinity) its kernel is the subspace defined by

the multiplicative subgroup schemes GZ dEZ (cf. 2.2) where we takeddef= pn,

n .

Proof. The morphism is what is usually referred to in p-adic Hodge theory asthe p-adic period map. There are many ways to construct the p-adic period map.Over the smooth locusMZ, one way to construct the period map (cf. [Mzk3],2,the Remark following the proof of Theorem 2.2) is to use the canonical section

(defined modulop

n

cf. [Mzk3], 1, the discussion following Lemma 1.1 for moredetails on this canonical section)

H :H (Z/pnZ) E (Z/pnZ)

of the universal extension E E (where E MZ is the tautological elliptic

curve) over the coveringH(def= E) EofEgiven by multiplication by pn (where

we let n ). By looking at fibers (ofH, E) over the origin ofE, we thussee thatHdetermines ahomomorphismfrom thep

n-torsion points ofE|TZ

, i.e., in

particular, fromF (Z/pnZ) E(TZ), toE (Z/pnZ)|

TZ, as desired. Note that

although only the smooth case is discussed in loc. cit., one verifies immediatelythat this approach may be extended over MZ (by using the canonical section ofthe universal extension near infinity). Lettingn and tensoring over Zp withO

TZthus completes the construction of.

The remaining statements concerning the various properties of may beverified by using another approach to constructing the p-adic period map, which isdiscussed in [Mzk1], Chapter IX, 2. This approach has the slight disadvantage that

it is not immediately clear that the resulting period map is defined overTZ (i.e.,a priori, it is only defined over locally constructed

Rs). Nevertheless, one may

check easily (by working over the ordinary locus) that these two approaches yieldthe same period map (cf. [Mzk3],2, the Remark following the proof of Theorem2.2). On the other hand, the approach of [Mzk1], Chapter IX, 2, has the advantagethat it makes it clear that the morphism : O

TZZp F E| TZ

in question is

the composite (at least locallyonTZ, but this is sufficient for our purposes) of amorphism which we shall denote 1 from O

TZZp F to a certain rank two

vector bundleonTZ, followed by a surjection which we shall denote2 fromthis vector bundle onto the line bundle E|

TZ(cf. the morphism Zp or,

more precisely, its dual of [Mzk1], Chapter IX, Theorem 2.5). Moreover, the

determinant of1 is equal to 1 p times a unit (cf. [Mzk1], Chapter IX, Theorem


24/28


2.5, as well as the proof of this theorem). On the other hand, since the rangeof the morphism is a line bundle, it follows that the image of determinesan idealJ O

TZwith the property that det(1) vanishes modulo J. Thus, we

conclude that 1 p J, as desired. Finally, the assertions of Theorem 3.1 overthe ordinary locus and near infinity follow from the theory of either [Mzk3],2, or[Mzk1], Chapter IX,2. This completes the proof

Remark. Usually, in the context ofp-adic Hodge theory, one constructs the p-adicperiod map not overglobal basessuch as MZ, but overp-adic bases, such as MZp . Itis immediate, however, from the first approach discussed in the proof of Theorem 3.1to constructing thep-adic period map thatexactly the same construction works overMZand gives rise to a morphism that isequivariant with respect toMQ =1(MQ)(i.e., not just 1(MQp)). (The author wishes to thank A. Tamagawa for remarksthat led to a simplification of the proof of Theorem 3.1.)

Remark. Thus, in particular, we obtain thatifEis a log elliptic curve overZ, then(by pull-back via the classifying morphism toM1,0 defined byE) the morphismdefines aGal(Q/Q)-equivariant morphism:

(E) :Tp(E) ZpO Q EOFO Q

(where Tp(E) is the p-adic Tate module ofE; Q is an algebraic closure ofQ; andthe hat denotes p-adic completion) whose image containsp E (cf. Theorems

3.1, 3.2). In particular, since the completions ofQ at each of its primes over pareall valuation rings, we thus conclude that (E) defines a Gal(Q/Q)-equivariantsurjection

(E) :Tp(E) ZpO Q E

(where EEZO Q is a free O Q-submodule of rank one) which coincides with

(E) over Qp. Thus, the kernel of(E)may be regarded as a global generalization

of the multiplicative subspace Zp(1) Tp(E) of the Tate curveE overZ[[q]].

Section 4: The Group Tensor Product

In this, we present a very general construction that applies to arbitrary com-mutative group schemes, but which will be of fundamental importance for motivat-ing the theory that we wish to pursue in subsequent papers.

We begin with the following definition, which is motivated by infinite abelian

group theory(cf. [Fchs], p. 94):


25/28


Definition 4.1. We shall call a torsion-free abelian groupRind-freeif every finitesubset ofR is contained in a finitely generated direct summand ofR.

Remark. It is known that every countable subgroup of an ind-free group is free(cf.

[Fchs], pp. 93-94). Thus, in particular, every countable ind-free group is free. Onthe other hand, groups such as an infinite direct product of infinite cyclic groupsare ind-free, but not free (cf. [Fchs], p. 94). The reason that we wish to considerind-free groups is because we wish to allow just such groups i.e., such as theadditive group of power series Z[[q]] in the theory to be developed in the presentand subsequent papers.

Now let S be a noetherian scheme, and G a commutative group schemeoverS. (Note: we do not necessarily assume that G is smooth or of locally of finite typeover S.) Thus,G defines a functor

T G(T)

from the category ofS-schemes (such asT S) to the category ofabelian groups.

Next, let us assume that we are given an ind-freeZ-moduleR (cf. Definition4.1). Then let us consider the functor

T G(T) ZR

which we denote byGgp R from the category ofS-schemes (such as TS)

to the category ofabelian groups.

Definition 4.2. We shall refer to as an ind-group scheme an inductive system(indexed by a filtered set) of group schemes in which the transition morphisms areall closed immersions.

Remark. Thus, a Barsotti-Tate group (e.g., the p-divisible group defined by an

abelian variety) is a typical example of an ind-group scheme. Note that it is im-portant to assume that the transition morphisms are closed immersions. Indeed,so long as one assumes this, taking the inverse limit of sheaves of functions on thegroup schemes in the system gives rise to a sheaf of functions on the ind-groupschemewhich surjects onto the sheaves of functions on each of the group schemesin the system. If, on the other hand, one considers an inductive system such as

EE EE E. . .

(where E is an elliptic curve, and all of the arrows are multiplication by some

positive integer d), then the inverse limit of the functions on the group schemes in


26/28


the system contains only the constant functions. Thus, it is difficult to treat thissort of inductive system as a single geometric object.

Proposition 4.3. This functorGgpRisrepresentableby an ind-group scheme

overS, which, by abuse of notation, we also denote byG gp R. Moreover:

(1) Ggp R isfunctorial with respect to G andR.

(2) If R is finitely generated, then Ggp R is representable by a

(single) group scheme.

(3) IfR andR are finitely generated freeZ-modules, andR R

is asplit injection, then the resulting arrowGgp

R Ggp

Ris a closed immersion.

Proof. First, note that any ind-freeR can be written as an inductive limit lim

Rj

offinitely generated freeZ-submodulesRj (for j in some filtered index set J) suchthat the injections Rj R are split. Moreover, when j < j

(so Rj Rj), thefact that Rj R is split implies that Rj Rj is split. Thus, the representability

ofGgpR(for a general ind-freeR) by an ind-group scheme over S follows formally

from assertions (1), (2), (3).

Let us prove (2), (3). Thus, for the remainder of this paragraph, let us assumethat R is finitely generated(and free). But thenR is (noncanonically) isomorphic

to Zr (for some positive integer r), so Ggp R can be represented by the group

scheme G SG S . . . SG (the fibered product over S of r copies ofG). This

proves (2). Since any split injection Zr Zr

is isomorphic to the injection Zr =

Zr {0} Zr Zrr = Zr

, assertion (3) follows immediately.

Finally, assertion (1) follows immediately from the functorial definition of

Ggp

R.

Definition 4.4. By abuse of notation, we denote the ind-group scheme of Propo-

sition 4.3 byGgpR, and refer to this ind-group scheme as the group tensor product

ofG withR.

Remark. We shall principally be interested in the case where G is an elliptic curveE, or its universal extension equipped with some integral structure e.g., E[d],et and R is a ring closely related to the base scheme over which E is defined

e.g., R= OF(the ring of integers) for some number fieldF.


27/28


Remark. Note that automorphisms of the module R induce automomorphisms of

Ggp R. Moreover, if, for instance,R is finitely generated, and G is equipped with

an ample line bundle L, then any choice of basis e1, . . . , er of R determines an

isomorphism Ggp R=

rj=1 G, hence (by taking the tensor product of the pull-

backs ofL relative to each of the factors in the product), a natural choice of ampleline bundle on G

gp R. This ample line bundle, however, depends on the choice

of basis, i.e., it will not be preserved in general by automorphisms ofR. Thus, insummary:

Although (for, say, finitely generatedR) the correspondence G

Ggp R defines a natural functor from group schemes to group

schemes, it doesnotdefine a natural functor frompolarizedgroupschemes to polarized group schemes.

Since, in Hodge-Arakelov theory, it is of fundamental importance to work withpolarized group schemes, the rather superficial general nonsense of the present willnot be sufficient for working with group tensored elliptic curves in Hodge-Arakelovtheory. The resolution of this technical issue forms one of the main obstacles relativeto developing a theory of the sort that the author envisions for globalizing the theoryof2.

One important property of the group tensor product is the following:

Proposition 4.5. Let d 1 be an integer, and write dG G (respectively,d(G

gp R) G

gp R) for the kernel of the morphism[d] :G G (respectively, [d] :

Ggp R G

gp R), i.e., multiplication byd. Then we have a natural isomorphism:

d(Ggp R)=(dG) Z/dZ(R/d R)

Proof. This follows immediately from the functorial definition ofGgp R.

Conclusion:

If we formally combine Proposition 4.5 with the content of the second Remarkfollowing Theorem 3.1, we thus obtain the following:

If E is a log elliptic curve overZ; d = pr (where p is a prime

number, and r 1 is an integer); and R def

= OK is the ring

of integers of some finite Galois extension K ofQ (which may


28/28


depend on the choice of d), then the rank 2 (R/dR)-module of

d-torsion points ofEgp Rcontains a rank1 submodule stabilized

byGal(Q/Q) and which coincides with the usual multiplicativesubspace at primes of bad (multiplicative) reduction.

In particular, this conclusion suggests that if we could somehow develop a Hodge-

Arakelov theory for objects such as Egp R, then we should be able to define for

such objects a Lagrangian Galois-theoretic Kodaira-Spencer morphism which is freeof Gaussian poles (cf. Corollary 2.5). In subsequent papers, it is the hope of theauthor to develop just such a theory.

Bibliography

[BO] P. Berthelot and A. Ogus,Notes on Crystalline Cohomology, Princeton Univ.Press (1978).

[MB] L. Moret-Bailly, Pinceaux de varietes abeliennes, Asterisque129, Soc. Math.France (1985).

[Fchs] L. Fuchs, Infinite Abelian Groups, Volume I, Pure and Applied Mathematics36, Academic Press (1970).

[Mzk1] S. Mochizuki, The Hodge-Arakelov Theory of Elliptic Curves: Global Discretiza-tion of Local Hodge Theories, RIMS Preprint Nos. 1255, 1256 (October 1999).

[Mzk2] S. Mochizuki, The Scheme-Theoretic Theta Convolution, RIMS Preprint No.1257 (October 1999).

[Mzk3] S. Mochizuki, Connections and Related Integral Structures on the UniversalExtension of an Elliptic Curve, RIMS Preprint No. 1279 (May 2000).

[Shi] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Forms,Publ. Math. Soc. of Japan 11, Iwanami Shoten and Princeton Univ. Press(1971).

The Galois-Theoretic Kodaira-Spencer Morphism of an Elliptic Curve

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