The Hodge-Arakelov Theory of Elliptic Curves in Positive Characteristic

8/13/2019 The Hodge-Arakelov Theory of Elliptic Curves in Positive Characteristic

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The Hodge-Arakelov Theory of Elliptic Curves

in Positive Characteristic

Shinichi Mochizuki

October 2000

Contents:

0. Introduction

1. The Verschiebung Morphism in Positive Characteristic

2. The Comparison Isomorphism in Positive Characteristic

3. Lagrangian Galois Actions in the 2-adic Case

3.1. Definition and Construction

3.2. Crystalline Properties

Section 0: Introduction

The purpose of this paper is to study the Hodge-Arakelov theory of ellipticcurves (cf. [Mzk1-4]) in positive characteristic. The first twos (1,2) are devotedto studying the relationship of the Frobenius and Verschiebungmorphisms of anelliptic curve in positive characteristic to the Hodge-Arakelov theory of ellipticcurves. We begin by deriving a Verschiebung-Theoretic Analogue of the Hodge-Arakelov Comparison Isomorphism(Theorem 1.1) which underlies our analysis in1,2. From this result, we derive, in particular, an explicit description of the etaleintegral structure of an elliptic curve in positive characteristic(Corollary 1.3). Thisresult may be regarded as a characteristic p version of [Mzk3], Theorem 2.2, which(unlike loc. cit., which holds only for ordinary elliptic curves) is valid even for

supersingular elliptic curves.Next, in 2, we apply the theory of 1 to obtain a new proof using posi-

tive characteristic methods (Theorem 2.3) of the scheme-theoretic portion of theHodge-Arakelov Comparison Isomorphism of [Mzk1]. In some sense, this new proofis more elegant than the proof of [Mzk1], which involves the verification of vari-ous complicated combinatorial identities (cf. the Remarks following Theorem 2.3).This situation is rather reminiscent of the computation of the degree of the hy-perbolically ordinary locus inp-adic Teichmuller theory([Mzk5], Chapter V cf.,especially, the second Remark following Corollary 1.3). Indeed, in that case, aswell, characteristic p methods (involving Frobenius and Verschiebung) give rise to

various nontrivial combinatorial identities. It would be interesting if this sort of

Typeset by AMS-TEX

1


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2 SHINICHI MOCHIZUKI

phenomenon could be understood more clearly at a conceptual level. Another in-teresting feature of the proof of Theorem 2.3 is the crucial use of a certain subgroupscheme of an elliptic curve in positive characteristic which may be regarded as ananalogue of the global multiplicative subspace of [Mzk4], 3. That is to say, thecrucial role of this subgroup scheme is reminiscent of the observation (cf. [Mzk4],

3,4) that such a global multiplicative subspace in the context of elliptic curves overnumber fields seems to be crucial to the application of Hodge-Arakelov theory todiophantine geometry.

Also, we remark that in the course of proving Theorem 2.3, we correct severalmisprints(cf. the Remark immediately following the proof of Proposition 2.2) in[Mzk3].

Finally, in 3, we work out the theory of [Mzk4], 2, in the case of p = 2.In loc. cit., this theory was only worked out in the case ofoddp (for the sake ofsimplicity). The case ofp = 2 involves dealing with various technical complications

modulo 2. Unlike the case of odd p, where the theory of [Mzk4], 2, allows one torelate the Lagrangian arithmetic Kodaira-Spencer morphism to the usual geometricKodaira-Spencer morphism of a family of elliptic curves, in the case ofp= 2, oneobtains the result (Corollary 3.7) that the Lagrangian arithmetic Kodaira-Spencermorphism is naturally related to the usual geometric Kodaira-Spencer morphism ofthe ample line bundle under consideration.

Notation and Conventions:

We will denote by (Mlog

ell)Z thelog moduli stack of log elliptic curves overZ (cf.[Mzk1], Chapter III, Definition 1.1), where the log structure is that defined by the

divisor at infinity. (In [Mzk1-4], Mell was denoted by M1,0. This change of nota-tion was adopted in response to the criticism voiced by a number of mathematicianswith respect to the notation M1,0.) The open substack of (Mell)Z parametrizing(smooth) elliptic curves will be denoted by (Mell)Z (Mell)Z.

Acknowledgements: The author would like to thank A. Tamagawa for stimulatingdiscussions of the various topics presented in this manuscript.

Section 1: The Verschiebung Morphism in Positive Characteristic

Fix a prime numberp. Let Slog be a fine noetherian log scheme over Fp, and

Clog Slog

alog elliptic curve(cf. [Mzk1], Chapter III, Definition 1.1) over Slog. WriteD Sfor the pull-back to Sof the divisor at infinity of the moduli stack of log ellipticcurves, and E C for the one-dimensional semi-abelian scheme which forms anopen subscheme of the semi-stable compactification C. Also, let us assume that

D S forms a Cartier divisorin S, and that on the open dense subscheme


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POSITIVE CHARACTERISTIC 3

USdef= S\D S

ofS, the log structure ofSlog is trivial.

Since S is an Fp-scheme, it is equipped with a Frobenius morphism

S :S S

Ifn is a nonnegative integer, then we shall denote the result of base-change withrespect to the n-th power of Sby means of a superscriptF

n. Note that the n-thpower of theVerschiebung morphism

VnE :EFn E

for the group scheme ESextends uniquely to a morphism

VnG :Gn C

satisfying the properties: (i) VnG |US =VnE|US ; (ii) in a neighborhood of the divisor

at infinity D, VnG is a finite etale covering of degree pn.

If the q-parameterof the log elliptic curve Clog Slog admits a pn-th rootatall points ofD, then let us write

Hn S

for the semi-stable genus 1 curve overSwhich is equal to E SoverUS, and, nearD, is the unique minimal semi-stable model ofE|US for which the closure of thepn-torsion points ofE|US lie in the smooth locus ofHn S. Then the morphism[pn]E :E Egiven by multiplication by p

n extends to a morphism

[pn]H :Hn C

which factors

HnnH Gn

VnG C

Moreover, this first morphism nH : Hn Gn may be identified with the n-thiterate of the relative (over S) Frobenius morphism Hn HF

n

n ofHn. Thus, inparticular, Gn may be identified with H

Fn

n . Also, over US, this factorization isthe usual factorization of [pn]E : E E as the composite of the n-th iterate ofFrobenius with the n-th iterate of the Verschiebung morphism.

Next, let us recall the universal extensionofE


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E E

(cf. [Mzk1], Chapter III, Definition 1.2), which extends naturally to an object

E

C C over C (cf. [Mzk1], Chapter III, Corollary 4.3). In [Mzk3], 1, weconstructed (in the case of a base which is flat over Z) an object EC,et C, i.e.,the universal extension equipped with the etale integral structure. By reducingthis object (in the case of a Z-flat base) modulo p, we may thus also speak of

EC,et S in the present context of a base S over Fp. Moreover, in addition tothis full etale integral structure, we also constructed intermediate etale integral

structures E;{N}C,et (where N 0 is an integer) which lie between E

C,et and E

C

and coincide withEC,et in relative degrees N cf. the discussion at the end of

[Mzk3], 4, for more details. Just as in the case ofEC,et, even though the E;{N}C,et

were defined over Z-flat bases, by reducing modulo p, it makes sense to speak of

the E;{N}C,et in the present context of a base Sover Fp.

Now let usfix an integern 0. Then it follows from the construction of theetale integral structure in [Mzk3], 1, that for somesufficiently large integerm n(depending on n), we have acanonical section

Hm :Hm E;{pn1}C,et

ofE;{pn1}C,et Cover the covering [p

m]H :Hm C. (Here, we assume, for the

moment that the q-parameter of our log elliptic curve admits a pm-th root at allpoints ofD. In fact, we shall see shortly that if suffices to take mequal to n.)

Thus, if we think of the structure sheaves of the various objects which are affineover C asOC-algebras, then pulling back functions by Hm defines a morphism

O


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of rankpn vector bundles onC. In particular, the morphismHm :Hm E;{pn1}C,et

factors as the triple composite of the morphism mH : Hm Gm, the naturalprojectionGm Gn, and a morphism

Gn :Gn E;{pn1}C,et

(which is necessarily unique).

Remark. In particular, in the simplest (nontrivial) case n = 1, we obtain an iso-morphism

O


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O


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hence (by applying the morphism PicQ(Gn) PicQ(C) given by taking norms ofline bundles) we have that:

[det(OGn)] = [det(OGn[VnG ])]|C

On the other hand, if we writeHn[nH]

def= HnGn 0Gn Hnfor thekernel of then-

th iterate of Frobenius(which is clearly a finite flat group scheme over S), then it iswell-known from the elementary theory of abelian varieties (cf., e.g., [Mumf4], 14,15) that the group schemes Hn[

nH] and Gn[V

nG ] are Cartier dualto one another,

hence that (as vector bundles over C) we have:

OHn[nH ]=(OGn[VnG ])

Moreover, since the Frobenius morphism is totally inseparable(so its fibers are all

geometrically connected) it follows immediately that

[det(OHn[nH ])] =pn1j=0

j [E]

hence that

[det(OGn[VnG ])] =[det(OHn[nH ])] =

pn1

j=0

j [E] = [det(O


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coefficient has the same p-adic absolute value as 1(pa)! . Then it is well-known and

easy to verify that the polynomialsTN

(forN < pn) may be written as polynomials

with Zp coefficients in the a(T), for a < n. Thus, it suffices to verify that

a(T+pn ) a(T) (mod pna)

for all Poly(Z,Zp), a < n.

We use induction on a. The assertion is clear for a = 0. If the assertion is truefor , a, then we have

a(T+pn ) = a(T) +pna

for some Poly(Z,Zp). Thus, it suffices to verify that

(a(T) +pna ) a+1(T) (mod pna1)

On the other hand, we have

(a(T) +pna ) =

a(T) +pna

p

=

p

j=0 a(T)

p j pna

j =

a(T)

p

+

pj=1

a(T)

p j

pna

j

a+1(T) (mod pna1)

(where we note thatpma

j

0 (mod pma1) for all j = 1, . . . , p). This com-

pletes the proof.

In particular, if, in Theorem 1.1, we assume S = US (for simplicity) and letn , then we obtain a characteristicpversion of [Mzk3], Theorem 2.2, which isvalid even for supersingular elliptic curves:

Corollary 1.3. (Explicit Description of theEtale Integral Structure ofan Elliptic Curve in Positive Characteristic) Assume thatS=US. Then theisomorphisms of Theorem 1.1 forn define an isomorphism:

Gdef= lim

nGn

Eet


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Hn Gn C

Hn

nH GnVnH C

(where both squares are cartesian, and we observe that the composite of the twolower horizontal arrows is the morphism [pn]H :Hn Carising from multiplication

bypn). Moreover, the sectionGn :Gn E;{pn1}C,et of Theorem 1.1 defines sections

Gn :Gn Gn; Hn :Hn H

n

that are compatible with each other and with the above commutative diagram.

Next, let us assume that we are given a torsion point

HHn(S)

oforderm, where (m, p) = 1. Write G Gn(S) for the image ofH in Gn(S).These torsion points define sheaves

LHdef= OHn(p

n [H]); LGdef= OGn([G])

on Hn and Gn, respectively, which are(ample) line bundles overUS. Near infinity,these sheaves are only coherent, but, in fact, may in most cases be treated asample line bundles, by working (in a neighborhood of infinity) with appropriatem-coverings

Hn Hn; Gn Gn

(i.e., coverings which induce bijections on the various irreducible components of thespecial fibers at infinity, and which induce the raising to the m-th power mapson each of the copies ofGm in the special fibers at infinity). These coverings havethe property that the divisors [H], [G] becomeCartierwhen pulled back via thesecoverings. Thus, one may think of sections of the sheaves LH, LG over Hn, Gn,as m-invariant sectionsof the resulting (ample) line bundlesover Hn, Gn. Note,moreover, that since m is a group scheme ofmultiplicative type, the operation oftaking m-invariants isexact.

In particular, it follows that (if, by abuse of notation, we denote all structuremorphisms to Sbyf) the push-forward sheaves

f(LH|Hn); f(LG|Gn)

admit natural filtrations Fr() (of ranks r pn, r, respectively, for r = 1, . . . , pn)

whose successive subquotientsFr+1/Fr() may be identified with


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rE OS f(LH); rE OS f(LG)

(where Edef= E), respectively. Moreover, since

nH is purely inseparable of degree

pn

, it follows that the divisor (n

H)1

([G]) is linearly equivalent to the divisorpn[H]. This gives rise to a natural Hn[nH]-action on LH (compatible with the

evidentHn[nH]-action on Hn itself). Relative to this action, we have

f(LG|Gn) =f(LH|Hn)Hn[

nH ]

(where the superscriptHn[nH] denotes Hn[

nH]-invariants).

We are now ready to define the evaluation mapsthat will appear in the com-parison isomorphisms. First, we observe that by using the sections Hn , Gn , we

may regard the group schemes

Hn[pn] Hn; Gn[V

nG ] Gn

(i.e., the kernels of the morphisms [pn]H : Hn C; VnG : Gn C, respectively)

as being contained in Hn, Gn, respectively. Thus, restriction to these subschemes

yields morphisms

H :f(LH|Hn) LH|Hn[pn]

G:f(LG|Gn) LG|Gn[VnG]

with the property that the latter morphism is the result of applying the operationof taking Hn[

nH]-invariantsto the former.

Before proving the comparison isomorphisms involving these evaluation maps,we would like to discuss some technical points, as follows: First, let us note thatthe tautological inclusion OGn OGn([G]) =LG defines a morphism

OSf(LG)

whose composite with the restriction morphismf(LG) LG|0G =OS(where weuse the fact that since the order m ofG isprime to p, we have 0G

G=) is the

identity. In fact, it is easy to see that here, 0G may be replayed by any pn-torsion

point, and hence that (by elementary algebraic geometry) we have the following:

Lemma 2.1. These two morphisms are isomorphisms, i.e., we have:

OS

f(LG)

LG| =OS


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(where Gn(S) is anypn-torsion point). In particular,f(LG) is a line bundleonSof degree zero.

Next, we would like to relate the evaluation maps constructed above to those

that appear in the theory of [Mzk1]. To do this, we would like first to make thefollowing observation concerning concerning integral structures over Z (or Z[12 ]):

Proposition 2.2. LetR def= Z ifp= 2, R

def= Z[12 ] ifp >2. Assume (just for the

remainder of this Proposition and its proof) thatSis a anR-flat scheme, and thatES is a family of elliptic curves overS. Notation:

(1) WriteE[pn]def= E E,[pn]E E (where [p

n]E :E Eis multipli-

cation by pn) cf. [Mzk3], 9. This object also has an etale

integral structure version E[pn],etdef= EetE,[pn]E E.

(2) Write def= OE([0E]); E

[pn] E for theHodge torsor asso-

ciated to the line bundleOE(pn [0E]) (cf. [Mzk3], 3) i.e.,

theE-torsor of connections on this line bundle. Note that thisis compatible with the notation of [Mzk3], 9. Moreover, there isan etale integral structure versionE[pn],et ofE

[pn] (cf. [Mzk3],

9, for details).

Then, if we think of these objects E[pn], E[pn],et, E[pn], E[pn],et as being variousR-integral structures on the objectE ZQ, then the followingcoincidences ofR-integral structures hold: E[pn] =E

[pn]; E

[pn],et =E

[pn],et.

Proof. Indeed, by working in the universal case (i.e., over (Mell)Z), one sees thatcoincidences of integral structures may be verified in a formal neighborhood ofinfinity, i.e., in the case where

S def= Spec(R[[q

12pn ]][q1])

and E S is the Tate curve with q-parameter equal to q. WriteE S for the

Tate curve withq-parameter equal toq 1pn . Then, if we think ofE(respectively,E)

as the quotient Gm/qZ (respectively, Gm/q

1pnZ ), then the natural inclusion

Z 1pn Z induces an etale isogeny

: E E

of degree pn (i.e., the morphism Gm/qZ Gm/q

1pnZ covered by the identity mor-

phism Gm Gm on Gm).


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Next, let us denote by E(S) the origin 0E(respectively, the 2-torsion point

defined by q 12pn ) ifp > 2 (respectively, ifp = 2). Then it follows easily from an

elementary computation namely, the fact that the sum of the fractions id

(for

i= 0, . . . , d 1) is equal to 12d d(d 1) = 12 (d 1) (which is Z ifd is odd, and

1

2Z

\Z

ifd is even) that

1([]) =pn [0E]

(where [] denotes the divisor class defined by the divisor inside the brackets).Let us denote the Hodge torsor associated to by (E) E. Then we have:

(E) EE=E[pn]

(by functoriality of formation of Hodge torsors) and

(E) E E=E[pn]

(cf. the discussion of [Mzk3], 9). Also, similar statements hold for etale integralstructure versions.

Now we are ready to compare integral structures. First, let us write Qfor the invariant (associated to and) denoted by i/2m in [Mzk3], 9. Sortingthrough the definitions, one verifies easily that ifp= 2 (respectively, p >2):

= 0 (respectively, =12 )

i.e., in either case, we obtain the key factthat R. If we split (E) E

via its canonical q-adic formal splitting, and write T for the usual coordinate onthe affine portion of (E) determined by the trivialization dU/U ofE (cf. thenotation of [Mzk3], 9), then the integral structure of (E) (respectively, (E)) isgiven by polynomials in:

T (respectively, T)

But since R, it follows that these two integral structurescoincide. The etaleintegral structure versions are handled similarly by considering

T

r

,Tr

instead

of (T )r, Tr (for r 0).

Remark. We would like to take this opportunity to correcttwo misprintsin [Mzk3],9, p. 78:

(i) The equation on the upper half of the page following the

phrase This makes it natural to define should read E[d],etdef=

Eet E,[d]E.


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(ii) In the first line of Definition 9.1, the phrase is to be shouldread to be.

Remark. Proposition 2.2 applies to the characteristicp discussionof the present

as follows: First of all, since the order m of H is prime to p, it follows that(at least over US) the Hodge torsor associated to LH is naturally isomorphic (cf.[Mzk3], Proposition 3.4) to (the reduction modulo p of) the objectE[pn] appearing

in Proposition 2.2. Thus, (since the object Hn|US of the present discussion is clearly

the same as the reduction modulo p of the object E[pn] appearing in Proposition

2.2) Proposition 2.2 implies that:

The Hodge torsor associated to LH is naturally isomorphic toHn|US .

A similar statement holds for etale integral structure versions. Thus, in particular,it follows that:

The evaluation morphism Hconstructed above may be identi-fied with the reduction modulo p of the evaluation morphism of[Mzk1], Introduction, Theorem A.

(cf. [Mzk3], 9, especially Theorem 9.2).

We are now ready to discuss the characteristic p approach to the Hodge-Arakelov Comparison Isomorphism of [Mzk1]:

Theorem 2.3. (Positive Characteristic Approach to the Hodge-ArakelovComparison Isomorphism) Letp be a prime number; n 1 an integer; and

Clog Slog

a log elliptic curve over a fine noetherian log scheme Slog in characteristic p.Write

Hn n

H Gn Vn

G C

for the factorization of the compactification [pn]H : Hn C of the morphismmultiplication bypn onE into then-th iterate of Frobenius, composed with then-th iterate of the Verschiebung morphism. Also, we introduce the notation:

C def= E

;{pn1}C,et ; G

n

def= C C,Vn

GGn; H

n

def= Gn Gn,nH Hn = C

C,[pn]H Hn

Moreover, let us assume that we are given atorsion point

HHn(S)


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oforder m, where (m, p) = 1. WriteG Gn(S) for the image ofH inGn(S),and

LHdef= OHn(p

n [H]); LGdef= OGn([G])

for the resulting sheaves on Hn, Gn. Then the section Gn : Gn E;{pn1}C,et of

Theorem 1.1 determines evaluation morphisms

H :f(LH|Hn) LH|Hn[pn]

G:f(LG|Gn) LG|Gn[VnG ]

with the following properties:

(1) G is an isomorphism overS.

(2) H is an isomorphism overUSdef= S\D.

Here, D S is the pull-back to S of the divisor at infinity of the moduli stack(Mell)Fp .

Proof. First, I claim that over US, G is an isomorphism if and only if H is.Indeed, to simplify notation, let us assume (just for the remainder of this paragraph)that US=S. Then it follows that the theta group (cf. [Mumf1,2,3]; [Mumf4], 23;or, alternatively, [Mzk1], Chapter IV, 1, for an exposition of the theory of thetagroups) GLH ofLH acts on the line bundle LH. Since LH has relative degree p

n

over S, this theta group fits into an exact sequence

1 Gm GLH E[pn] 1

and the fact that (nG)LG = LH implies that LG determines a section sG :

E[nE] GLH of this exact sequence over E[nE] E[p

n] E. In the termi-nology of theta groups, the image ofsGis aLagrangian subgroupof the theta group(cf. [MB], Chapitre V, Definition 2.5.1). It thus follows from the theory ofGLH -

modules that the GLH -linear morphism H is an isomorphism if and only if themorphism G which is obtained from H by taking E[nE]-invariants is anisomorphism. This completes the proof of the claim.

The fact that G is an isomorphism may be proven by arguing as follows:First, we observe that it suffices to work in the universal case, where, say, S isproper, connected, and smooth of dimension1 over Fp, and the classifying morphism

S (Mell)Fp is finite. Then Gis a morphism between two vector bundles of rankpn on S, hence it will be an isomorphism as soon as we verify the following twofacts:

(1) G is an isomorphism over the generic point ofS.


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(2) The degrees of the domain and range of G coincide.

Fact (1) follows from the second Remark following Proposition 2.2, together withthe analysis in a neighborhood of infinitygiven, for instance, in [Mzk1], ChapterV, Theorem 6.2 (cf. also [Mzk3],6). Fact (2) follows from the fact that (in light

of Lemma 2.1) the domain (respectively, range) of G has the same degree as thedomain (respectively, range) of the isomorphism of Corollary 1.4. This completesthe proof that G is an isomorphism.

Remark. Thus, in particular, the above argument gives anew proof of the (scheme-theoretic) characteristic zero portion of [Mzk1], Theorem A at least in the casewhend is a power of a prime number, and mis prime tod. More precisely, althoughwe used the computation at infinity of [Mzk1], the characteristic p argument givenabove may be used to replace the complicated degree computations(especially whend is even!) of [Mzk1], Chapter VI, proof of Theorem 3.1. (Note that although

here we are working in characteristicp, we obtain characteristic zero consequences,since any morphism between vector bundles on a flat, proper Z-scheme which isan isomorphism modulo pis necessarily an isomorphism over Q.) Also, we observethat, in fact:

The above argument furnishes a new proof of the scheme-theoretic,characteristic zero portion of the Hodge-Arakelov ComparisonIsomorphism (i.e., [Mzk1], Theorem A) for arbitrary d, m (asin the statement of this Theorem).

Indeed, this follows from the fact that the essential point of this characteristiczero portion of the theorem is a certain coincidence of degrees (cf. the degreecomputations of [Mzk1], Chapter VI, proof of Theorem 3.1). On the other hand,it is relatively easy to see (without computing the degrees precisely!) that the twodegrees in question are bothpolynomials ind. Thus, their difference is a polynomialin d which vanishes (by the above characteristic p argument) for all d equal to apower of (sufficiently large) p. But this implies that this difference is identicallyzero(for all d).

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

The characteristic p methods (involving the Frobenius and Ver-schiebung morphisms) of the present paper yield a new proof ofthe various combinatorial identities inherent in the computationof degrees in [Mzk1], Chapter VI, proof of Theorem 3.1.

This situation is rather reminiscent of the situation of [Mzk5], 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 of

phenomenon could be understood more clearly at a conceptual level.


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Remark. One interesting feature of the above proof is the crucial use of the isogenynH : Hn Gn, i.e., (over US) the (n-th iterate of the) Frobenius morphism

nE :

E EFn

. Put another way, this amounts to the use of the subgroup schemeE[nE]E(i.e., the kernel of

nE), which, of course, does not exist in characteristic

zero. Note that this subgroup scheme is essentially the same as the multiplicative

subspacethat played an essential role in [Mzk4], 2. That is to say, it is interestingto note that just as in the context of [Mzk4] (cf., especially, 3, 4) the crucialarithmetic object that one wants over a number field is a global multiplicativesubspace, in the above proof, the crucial arithmetic object that makes the proofwork (inpositive!characteristic) is the global multiplicative subspaceE[nE] E(which is defined over all of (Mell)Fp).

Remark. 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 flatgroup scheme on a proper curve always has degree zero):

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

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 Gaussian poles(cf., e.g., [Mzk1], Introduction, 1).

Section 3: Lagrangian Galois Actions in the 2-adic Case

In [Mzk4],2, we assumed (for the sake of simplicity) that the prime of interestp wasodd. In the present, 2, we would like to work out the theory of [Mzk4], 2,in the case p = 2. This involves dealing with various subtle technical issues modulo2.

3.1. Definition and Construction

Let p= 2. Let d >1 be a power of 2. Let A be a complete discrete valuation

ring of mixed characteristic(0, p), with perfect residue field, whichcontains all the2d-th roots of unity. WriteK(respectively, k) for the quotient field(respectively,residue field) ofA.

Set

S def= Spec(A[[q

1N ]])

for some oddpositive integer N. Endow S with the log structuredefined by thedivisor V(q

1N ) S, and denote the open subscheme ofSwhere the log structure

is trivial by USS. Write


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Sdef= 1(US Q)

(for some choice of basepoint), and

Clog Slog

for the log elliptic curve determined by the Tate curve, i.e., the degeneratingelliptic curve E S (more precisely: one-dimensional semi-abelian scheme) withq-parameter equal to q OS.

Set

Z def= Spec(A[[q

12Nd ]])

Endow Z with the log structuredefined by the divisor V(q 12Nd ) Z. Thus, we

obtain a morphismZlog Slog of log schemes.

Next, let us write

Ed,Z Z

for the object which is equal to the one-dimensional semi-abelian scheme EZdef=

ESZ over UZdef= USSZ, and, near infinity, is the pull-back to Z of the

object Ed (cf. [Mzk1], Chapter IV, 4, where we take N of loc. cit. to bed). In words, this object Ed is the result of removing the nodes from the uniqueregular semi-stable model of the Tate curve (with q-parameter q) over the base

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

E[d],et,ZEd,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.

Next, let us observeEZhas aunique finite flat subgroup schemeGZannihilated

byd. This subgroup scheme is naturally isomorphic to d. Thus, we have:

d=G

ZEZ[d] Ed,Z

(where EZ[d] Ed,Zdenotes the closed subscheme which is the kernel of multipli-cation by d on Ed,Z). Note, moreover, that, in fact,G

Zdescends to a subscheme

GSE[d]E over S.


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Since the quotient (EZ[d])/GZ is naturally isomorphic to the constant group

scheme (Z/dZ)Z, it is easy to see that, over Z, there exists a finite etale groupschemeHZEZ[d] such that the natural morphism

G

ZZHZEZ[d]

is an isomorphism of group schemes. Thus, if we writeEHZdef= Ed,Z/HZ, then we

see that EHZ Z is a one-dimensional semi-abelian group scheme(i.e., its fibersare all geometrically connected), and that the natural quotient morphism

(EZ) Ed,Z EHZ

(over Z) has kernel equal to HZ, hence is finite etale of degree d. Moreover, wenote that the q-parameter ofEHZ is a d-th root of q. In particular, (unlike G

Z

)HZ is notdefined over S.

Next, let us assume that we have been given an odd integerm > 1, togetherwith a torsion point

E,S(S)

of order preciselymwhich defines a metrized line bundle

L

def

= Lst,

on E,S (cf. [Mzk1], Chapter V, 1, for a discussion of the construction of theobject Lst,). Thus, in particular, over US:

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

Here, we recall thatSis the stack(in thefinite, flat topology) obtained fromSbygluing togetherUS(away from infinity) to the profinite covering ofS(near infin-ity) defined by adjoining a compatible system ofM-th roots of theq-parameter

(as M ranges multiplicatively over the positive integers). Over S, we have thegroup 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.

Note that L has an associated theta group (cf. [Mzk1], Chapter IV, 1, 5, for

a discussion of theta groups) GZ over Zwhich fits into an exact sequence:


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1 (Gm)Z GZEZ[d] 1

Also, let us assume that we are given a lifting

HZ GZ

ofHZ (i.e., HZHZ viaGZ EZ[d]). Thus,HZ is a Lagrangian subgroup (cf.

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

Remark. To see that such a lifting HZ GZexists, one may, for instance, apply thecanonical sectionof [Mzk1], Chapter IV, Theorem 1.6, (1), over a double coveringEd,Z Ed,Z (so L|Ed,Z will be a (metrized) line bundle of degree 2d) with the

property that HZEd,Zlifts to a subgroup scheme HZE

d,Zsuch thatH

Z

HZ.Note that since theq-parameter ofEd,Zwill then necessarily be a square root ofq,

it follows that in order to ensure thatHZexist, we need to know the existence of a2d-th rootofq inOZ. This is why we defined Z as we did (i.e., rather than withthe 2N d replaced by N d, as was done in the case ofoddp).

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 toE[d],et,Z. Here, we take the

integral structure of this push-forward near infinity to be the unique GZ-stableintegral structure determined by the CGr cf. [Mzk1], Chapter V, Theorem4.8; the discussion of [Mzk3], 4.1, 4.2. Thus,VL is a quasi-coherent sheaf onZ,equipped with a filtration Fr(VL) VL, i.e., the subsheaf consisting of sectionswhose torsorial degree is< r. (Here, by torsorial degree, we mean the relativedegree with respect to the structure of relative polynomial algebra on OE overOE (arising from the fact that E

E is an affine torsor). Since E may beidentified with E[d],et,Z over (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.

Now we come to the portion of the discussion involving phenomena unique to

the primep= 2.


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Proposition 3.1. The objects introduced above satisfy the fol lowing properties:

(i) The integral structure ofE[d],et,Zis equal to that ofE[d],et,Z

def=

E[d],et|Z(i.e., the pull-back via the multiplication byd morphism

Ed,Z EZ of the universal extension Eet|Z over Z equippedwith the etale integral structure cf. the notation of [Mzk3],9). In particular, thed-torsion subgroup schemeEZ[d]Ed,Zlifts naturally to a subgroup schemeEZ[d] E

[d],et,Z.

(ii) The canonical section

(EZ[d] GZ) 2 G

Z GZ

(cf. [Mzk1], Chapter IV, Theorem 1.6, (1)) extends to all ofGZ.In particular, (despite the fact thatd is even!) we obtain atheta

trivializationL|G

Z

=L|0EZ OZ OGZ

Here, 0EZ EZ(Z) is the zero section ofEZdef= ESZZ.

Proof. Assertion (i) follows from the fact that the integral structure used to define

E[d],et,Z is given by d(T(i/2m))

r

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

which gives the same integral structure as

dTr

. Note that here we use the

assumptions that (a.) mis odd; (b.) d >1 is even.Assertion (ii) is proven by observing that, if we descend L to some LH (via

the lifting HZ GZ discussed above) on EHZ , the resulting degree 1 (metrized)line bundle is (up to translation by an odd order torsion point) that defined by anonmultiplicative(i.e., lying outside the image ofGZ inEHZ )order2 torsion point.

In particular, the invariant i/2m associated to this LH is Z2 (cf. the theoryof [Mzk1], Chapter V,4; as well as [Mzk1], Chapter IV, Lemma 5.4; the discussionof [Mzk3], 4.3). Put another way, the essential phenomenon at work here is theelementary numerical fact that (for D 1 an integer)

1

D

D1j=0

j = 1

2(D 1)

lies Z ifD isodd, and 12Z\Z ifD is even.

On the other hand, (as one may recall from the discussion of [Mzk3], 4.3 cf., especially, the proof of Lemma 4.1):

The class of this invariant i/2m associated to LH in 12Z2/Z2

is precisely the obstruction to the existence of the desired section

GZ GZ.


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Thus, the fact that this invariant is Z2 implies that this obstruction is 0, asdesired.

Remark. The description ofLHgiven in the above proof shows that in fact, this (apriori) metrized line bundle is defined as an ordinary line bundleover some semi-stable model ofEHZ = Ed,Z/HZ over Z. (More precisely, the semi-stable modelwith the property that the group of irreducible components of its special fiber isequal to 12NZ/Z is sufficient.) In particular, it follows thatL (respectively, VL) isdefined as an ordinary line bundle (respectively, ordinary vector bundle) oversome semi-stable model ofEZ Z(respectively, over Z).

Next, let us note that since L is defined overE,S (i.e., without base-changingto Z), it follows that L|0EZ is, in fact, defined over S (i.e., in other words, it isdefined over S, except that near infinity, one may need to adjoin roots 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 .

Let us denote HZ-invariantsby means of a superscript HZ. Then recall thatVHZL

admits the following interpretation: Since HZ = HZ acts on Ed,Z; E,S;

E[d],et,Z; L, we may form the quotientsof these objects by this action. This yields

objects (Ed,Z)H (i.e., EHZ ), (E,S)H, (E[d],et,Z)H, LH(a metrized line bundle on

(E,S)H). Then we have:

VHZL

=f{LH|(E[d],et,Z

)H}

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

Thus, by restricting HZ-invariant sections ofL over E[d],et,Z i.e., sections

ofLHover (E[d],et,Z)H to G

ZEZ[d]

=EZ[d] E[d],et,Z, and composing with

the theta trivialization of Proposition 3.1, (ii), we obtain a morphism:

HZV : VHZL

L|0EZ OZ OG

Z

Similarly, if we introduce Gaussian poles (cf. [Mzk1], Introduction, Theorem A,

(3); [Mzk3], Theorem 6.2), we get a morphism:

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

Z

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

Corollary 3.2. (Lagrangian Version of the Main Result of [Mzk1])Assume that d is a power of p = 2, and thatm isodd. Then restriction ofHZ-invariant sections ofVL to G

Zgives rise to a morphism

HZV : V

HZL L|0EZ OZ OGZ


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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 Corollary is a special case of [Mzk1], Introduction, Theorem A, (2),(3). Note that the zero locus of the determinant is empty because of our assump-tion that m isodd(hence invertible on S).

Definition 3.3. The natural action of S

onG

Z, together with the isomorphism

GP,HZH of Corollary 2.2, and the natural action of S on L|0EZ , define a natural

action of S on HGP,HZDR , which we shall refer to as the Lagrangian Galois action

onHGP,HZDR .

Remark. Just as in [Mzk4], 2.1, (unlike the naive Galois action) the LagrangianGalois actiondepends on the choice of the additional dataGZ, HZ. Also, just as in[Mzk4],2.1, althougha priori, the Lagrangian Galois action appears to require theGaussian poles(i.e., it appears that it is not necessarily integrallydefined onHHZDR),in fact, however, we shall see in 3.2 below that the Lagrangian Galois action has

the remarkable property that it is defined without introducing the Gaussian poles.

3.2. Crystalline Properties

We maintain the notation of3.1.

The first portion of [Mzk4], 2.2, now goes through with little change: As inloc.cit., we first would like torelate the present discussion to the theory of connectionsin [Mzk3]. Thus, recall that L|0EZ is a line bundle on Zequipped with a naturalS-actionderived from a trivialization

:L|0EZ=q

a2Nd OZ

(where a is a nonnegative integer < 2N d) which, in the terminology of thediscussion of [Mzk3], 5, determines a S-equivariant rigidification ofL at 0EZ .In particular, defines a S-invariant logarithmic connectionon the line bundleL|0EZ . Thus, by the theory of [Mzk3], 5, this rigidification gives rise to a S-invariant (logarithmic) connection

VHZ

L


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on VHZL

(cf. [Mzk3], Theorems 5.2, 8.1). Here, the logarithmic connections are

relative to the log structure ofZlog, and all connections, differentials, etc., are tobe understood as being continuous with respect to the (p, q)-adic topology on OZ.

Just as in [Mzk4], 2.2, sinceall higherp-curvatures of these connections vanish

(cf. [Mzk3], 7.1, for a discussion of the general theory of higher p-curvatures;[Mzk3], Corollary 7.6, for the vanishing result just quoted), we thus conclude thatthe pair

(VHZL

, VHZ

L

)

defines a crystalon the site

Inf(Zlog k/A)

of (all i.e., not just PD-)infinitesimal thickenings overA of open sub-log schemesofZlog k= Zlog (A/mA).

One verifies immediately (using the simple explicit structures ofS,Z) that theaction of S onOZ satisfies:

() (mod mA OZ)

S, OZ, and that the correspondence

S (q 12Nd )/q

12Nd

defines morphisms:

S Gal(Z/S) (Z/2dZ)(1)

(where the first (respectively, second) arrow is a surjection (respectively, isomor-phism)).

Next, let us observe that the property just discussed concerning the action ofS onOZimplies that every Sdefines 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

(ii) the fact that (V

HZ

L , VHZ

L ) forms a crystal on Inf(Z

log

k/A)


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that induces a -semi-linear isomorphism

:VHZL

VHZL

(where -semi-linear means semi-linear with respect to the action of on OZ,and the hat denotes p-adic completion).

As in [Mzk4], 2.2, the justification for the notation

is that this iso-morphism is the analogue of the isomorphism obtained in differential geometry by

parallel transporting i.e., integrating sections ofVHZL

along the path

(where we think of as an element of the (algebraic) fundamental group S).

That is to say, we obtain a naturalS-semi-linear action ofS onVHZL .Theorem 3.4. (Crystalline Nature of the Lagrangian Galois Action)

The action ofSonVHZL is compatible withHZV (cf. Corollary 3.2; here the hatdenotes p-adic completion) and the natural action of S on G

Z in the following

sense: For S, the following diagram commutes:

VHZL

HZ

V L|0EZ OZ OGZ

VHZL

HZ

V L|0EZ OZ OGZ

(where the on the right denotes the result of applying to OGZ

via the natural

action ofS onOGZ

).

Proof. As in [Mzk4], 2.2, this follows from the naturality of all the morphismsinvolved, together with the compatibility (cf. [Mzk3], Theorem 6.1) over GZof theconnection

VHZ

L

with the theta trivialization of Proposition 3.1, (ii).

Corollary 3.5. (Absence of Gaussian Poles in the Lagrangian GaloisAction) Relative to the objects of the present discussion, the Lagrangian Galois

action ofS onHGP,HZ

DR

(cf. Definition 3.3) is definedwithout Gaussian poles,

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

Proof. As in [Mzk4],2.2, this follows formally from the commutative diagram ofTheorem 3.4, together with Lemma 3.6 below.

Lemma 3.6. The image of the morphismHZV

(cf. Corollary 3.2) is the same

as the image of its restrictionHZH to HHZDR

VHZL

.

Proof. The proof is entirely similar to that of [Mzk4],2.2, Lemma 2.6.


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Finally, just as in [Mzk4], 2.2, we observe that:

Theorem 3.4 allows us to relate the arithmetic Kodaira-Spencermorphism arising from the Lagrangian Galois action to the clas-

sical geometric Kodaira-Spencer morphism.

The argument in the case p= 2, however,differs somewhatfrom the case of oddp.

We begin as in [Mzk4], 2.2. Let

Gal(Z/S)

be asubgroup of order> 1. Writeddef= ||for the order of . Thus, d= 1 divides

2d, and we have a natural isomorphism=(Z/dZ)(1). 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) (which

is 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/m

2.

Next, as in [Mzk4], 2.2, we would like to consider a certain crucial portionof the arithmetic Kodaira-Spencer morphism associated to the Lagrangian Galois

action. Ultimately, however, this crucial portion in the case of p = 2 will differsomewhat from the crucial portion in the case ofpodd.

Let . Then since acts on HHZDR via the Lagrangian Galois action

(Definition 3.3, Corollary 3.5), we see that defines a morphism HHZDR HHZDR. If

we restrict this morphism to F1(HHZDR), and compose with the surjectionHHZDR

{HHZDR/F2(HHZDR)} A(A/m

2), we thus obtain a morphism

F1(HHZDR) {HHZDR/F

2(HHZDR)} A(A/m2)

which, as in [Mzk4],2.2, defines a morphism


27/29


28/29


LH : (Zlog/A) EHZ

(i.e., the morphism that describes the variation in the moduli of this line bundle)on

log(q1

2Nd )

(Zlog/A) and (as in [Mzk4], 2.2) multiplying the result by .

Moreover, since the line bundle LH is defined by some torsion point (cf. theproof of Proposition 3.1, together with the Remark following this proposition), i.e.,

if we think ofEHZ as Gm/q1dZ, the point defined by some

qa1

2a2d

(where a1, a2 Z are relatively prime and odd), we conclude that the effect of theaction ofon the moduli ofLHis given by

(q 2d ) q

2d + ( N) () q

2d q

2d {1 + ()} (mod m

2)

(where def= a1/a2, and, in the second congruence, we use the fact that a1, a2, and

Nare odd).

We summarize this discussion as follows:

Corollary 3.7. (Relation to the Classical Geometric Kodaira-SpencerMorphism) Let Gal(Z/S)

be asubgroup of order >1. 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 12Nd modulo m2). Then the morphism

: HomOZ

(F1(HHZDR

) k, (F2/F1)(HHZDR

) m

/m2

) =EHZ

m

/m2

obtained purely from theLagrangian Galois action of onHHZDR (cf. Definition

3.3, Corollary 3.5) by restricting this action to F1(HHZDR)and then reducing modulom2coincideswith the morphism obtained by evaluating thegeometric Kodaira-

Spencer morphism of the line bundle LH on EHZ

LH : (Zlog/A) EHZ

in the logarithmic tangent direction log(q

12Nd )

(Zlog/A) and multiplying the

result by.


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Moreover, if we trivialize(Zlog/A) via this logarithmic tangent direction and

EHZ via the logarithmic tangent direction U (whereU is the standard multiplica-

tive coordinate on the copy ofGm that naturally uniformizesEHZ ), thenLH is theidentity morphism modulo m.

Thus, in summary, thearithmetic Kodaira-Spencer morphismassociatedto the Lagrangian Galois action in the case p = 2 coincides modulo m2 with theusual geometric Kodaira-Spencer morphism of the ample line bundleunder consideration.

Remark. Just as in [Mzk4], 2.2, the correspondence between the logarithmic tan-gent direction

log(q1

2Nd )(Zlog/A)

and the morphism is essentially the same

as the correspondence arising from Faltings theory of almost etale extensions be-tween the logarithmic tangent bundle of Zlog and a certain Galois cohomology

group (cf., e.g., [Mzk1], Chapter IX, 2, especially Theorem 2.6, for more details).

Bibliography

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

[Mumf1,2,3] D. Mumford, On the equations defining abelian varieties I, II, III,Invent. Math.1 (1966), pp. 287-354; 2 (1967), pp. 71-135; 3 (1967), pp. 215-244.

[Mumf4] D. Mumford,Abelian Varieties, Oxford Univ. Press (1974).

[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).

[Mzk4] S. Mochizuki, The Galois-Theoretic Kodaira-Spencer Morphism of an EllipticCurve, RIMS Preprint No. 1287 (July 2000).

[Mzk5] S. Mochizuki, Foundations of p-adic Teichmuller Theory, AMS/IP Studiesin Advanced Mathematics 11, American Mathematical Society/InternationalPress (1999).

The Hodge-Arakelov Theory of Elliptic Curves in Positive Characteristic

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