Advanced Studies in Pure Mathematics2*,2000
Analysis on Homogeneous Spaces and Reprentations of Lie Groups
pp.1–34
A Survey of the Hodge-Arakelov Theory of猪寄生虫病
Elliptic Curves II
Shinichi Mochizuki1
Abstract.
The purpo of the prent manuscript is to continue the survey of the Hodge-Arakelov theory of elliptic curves(cf.[7,8,9,10,11])
义工活动that was begun in[12].This theory is a sort of“Hodge theory of ellip-
tic curves”analogous to the classical complex and p-adic Hodge the-
ories,but which exists in the global arithmetic framework of Arakelov
theory.In particular,in the prent manuscript,we focus on the as-泡泡宝盒
pects of the theory(cf.[9,10,11])developed subquent to tho
discusd in[12],but prior to the conference“Algebraic Geometry
2000”held in Nagano,Japan,in July2000.The developments
center around the natural connection that exists on the pair con-
sisting of the universal extension of an elliptic curve,equipped with
an ample line bundle.This connection gives ri to a natural ob-
ject—which we call the crystalline theta object—which exhibits
many interesting and unexpected properties.The properties al-
low one,in particular,to understand at a rigorous mathematical
level the(hitherto purely“philosophical”)relationship between the
classical Kodaira-Spencer morphism and the Galois-theoretic“arith-
metic Kodaira-Spencer morphism”of Hodge-Arakelov theory.They
also provide a method(under certain conditions)for“eliminating the
Gaussian poles,”which are the main obstruction to applying Hodge-
Arakelov theory to diophantine geometry.Finally,the techniques
allow one to give a new proof of the main result of[7]using charac-
teristic p methods.It is the hope of the author to survey more recent
,developments that occurred subquent to“Alge-
braic Geometry2000”)concerning the relationship between Hodge-
Arakelov theory and anabelian geometry(cf.[16])in a quel to the
prent manuscript.
Received January31,2001.找的成语
Revid January31,2001.
1Part of the rearch discusd in this manuscript was carried out while the author was visiting the University of Tokyo during the Spring of2000as a“Clay Prize Fellow”supported by the Clay Mathematical Institute.
2S.Mochizuki
Contents
1.General Introduction2
2.The Crystalline Theta Object5
3.Lagrangian Galois Actions20
4.Hodge-Arakelov Theory in Positive Characteristic28
§1.General Introduction
四川名菜We begin our general introduction to the topics prented in the prent manuscript by reviewing the fundamental argument from al-gebraic geometry who arithmetic analogue is the central goal of the Hodge-Arakelov theory of elliptic curves.
Let S be a smooth,proper,geometrically connected algebraic curve over an algebraically clodfield of characteristic zero k.Let
E→S
be a family of one-dimensional mi-abelian varieties who genericfiber is proper.Thus,(except for afinite number of exceptions)thefibers of E→S are elliptic curves.Let us write
Σ⊆S
for thefinite t of points over which thefiber of E→S fails to be an elliptic curve,and
ωE def=ΩE/S|0E
for the restriction of the sheaf of relative differentials to the zero ction of E→S.Then the height of the family E→S is defined to be:
ht E def=deg S(ω⊗2
)
E
(i.e.,the degree on S of the line bundle in parenthes).The height is a measure of the arithmetic complexity of the family E→S.For instance, the family is ,becomes trivial upon applying somefinite flat ba extension T→S)if and only if ht E=0.
In some n,the most important property of the height in this context is the fact that(in the nonisotrivial ca)it is universally bounded by invariants depending only on the pair(S,Σ).This bound—“Szpiro’s conjecture for functionfields”—is as follows:
Survey of Hodge-Arakelov Theory II3
(1)ht E≤2g S−2+|Σ|
(where g S is the genus of S,and|Σ|is the cardinality ofΣ).The proof,in the prent geometric context,is the following simple argument:Write M ell for the compactified moduli stack of elliptic curves over k,and ∞⊆M ell for the divisor at infinity of this stack.Thus,E→S defines
a classifying morphism
κ:S→M ell
who(logarithmic)derivative
dκ:ω⊗2
E ∼=κ∗Ω
M ell/k
(∞)→ΩS(Σ)
is nonzero(so long as we assume that the family E→S is nonisotrivial). Thus,since dκis a generically nonzero morphism between line bundles on the curve S,the degree of its ,ht E)is≤the degree of its ,2g S−2+|Σ|),so we obtain the desired inequality.
Note that in the above argument,the most esntial ingredient is the Kodaira-Spencer ,the derivative dκ.Until recently, no analogue of such a derivative existed in the“arithmetic ca”(i.e.,of elliptic curves over numberfields).On the other hand:
The Hodge-Arakelov theory of elliptic curves gives ri
to a natural analogue of the Kodaira-Spencer morphism
in the arithmetic context of an elliptic curve over a
numberfield.
A survey of the basic theory of this arithmetic Kodaira-Spencer mor-phism,together with a detailed explanation of the n in which it may be regarded as being analogous to the classical geometric Kodaira-Spencer morphism,may be found in[12].
At a more technical level,in some n the most fundamental result of Hodge-Arakelov theory is the following:Let E be an elliptic curve over afield K of characteristic zero.Let d be a positive integer,and η∈E(K)a torsion point of order not dividing d.Write
L def=O E(d·[η])
for the line bundle on E corresponding to the divisor of multiplicity d with support at the pointη.Write
E†→E
4S.Mochizuki
for the universal extension of the elliptic ,the moduli space of pairs(M,∇M)consisting of a degree zero line bundle M on E,equipped with a connection∇M.Thus,E†is an affine torsor on E under the moduleωE of invariant differentials on E.In particular,since E†is (Zariski locally over E)the spectrum of a polynomial algebra in one variable with coefficients in the sheaf of functions on E,it makes n to speak of the“relative degree over E”–which we refer to in this paper as the torsorial degree–of a function on E†.Note that(since we are in characteristic zero)the subscheme E†[d]⊆E†of d-torsion points of E†maps isomorphically to the subscheme E[d]⊆E of d-torsion points of E. Then in its simplest form,the main theorem of[7]states the following: Theorem 1.1.(Simple Version of the Hodge-Arakelov Com-parison Isomorphism)Let E be an elliptic curve over afield K of characteristic zero.Write E†→E for its universal extension.Let d be a positive integer,andη∈E(K)a torsion point who order does not
divide d.Write L def=O E(d·[η]).Then the natural map
Γ(E†,L)<d→L|E†[d]圣淘沙花园
given by restricting ctions of L over E†who torsorial degree is<d to the d-torsion points E†[d]⊆E
†is a bijection between K-vector spaces of dimension d2.
The remainder of the main theorem esntially consists of specifying precily how one must modify the integral structure ofΓ(E†,L)<d over more general bas in order to obtain an isomorphism at thefinite and infinite primes of a numberfield,as well as for degenerating elliptic curves.
The relationship between Theorem1.1and the classical Kodaira-Spencer morphism is discusd in detail—albeit at a rather philosophi-cal level—in[12],§1.3,1.4.The analogue in the arithmetic ca of the geometric argument ud above to prove(1)is discusd in[12],§1.5.1. The upshot of this argument in the arithmetic ca is that in order to derive diophantine equalities analogous to(1)in the arithmetic ca—i.e.,Szpiro’s conjecture—it is necessary to eliminate certain unwanted poles—called Gaussian poles—that occur in the construction of the arithmetic Kodaira-Spencer morphism.
In the prent manuscript,we discuss the following further develop-ments in the theory(cf.§3,4):
Survey of Hodge-Arakelov Theory II5
(i)a method for eliminating the Gaussian poles under certain con-
ditions(cf.Corollary3.5);
(ii)an argument which shows that(under certain conditions)the reduction in positive characteristic of the arithmetic Kodaira-
Spencer morphism coincides with the classical geometric Kodaira-
Spencer morphism(cf.Corollary3.6);
(iii)a alternative proof of Theorem1.1using characteristic p meth-ods(cf.Theorem4.3).
Thus,development(i)brings us clor to the goal of applying Hodge-Arakelov theory to proving Szpiro’s conjecture(for elliptic curves over numberfields).Unfortunately,the conditions under which the argu-ment of(i)may be carried out do not hold(at least in the naive n) for elliptic curves over numberfields.Nevertheless,there is substan-tial hope that certain new constructions will allow us to realize the conditions even for elliptic curves over numberfields(cf.§3for more on this issue).Development(ii)is significant in that it shows that the analogy between the arithmetic and classical geometric Kodaira-Spencer morphisms is not just philosophy,but rigorous mathematics!Finally, the significance of development(iii)is that it provides a much more con-ceptual,as well as technically simpler proof of the“fundamental theorem of Hodge-Arakelov theory”(i.e.,Theorem1.1).
Underlying all of the new developments(especially(i),(ii))is the theory of the crystalline theta object,to be discusd in§2.This object
is a locally free sheaf equipped with a connection and a Hodgefiltration, hence is reminiscent of the“MF∇-objects”of[2],§2.On the other hand,many of its properties,such as“Griffiths mi-transversality”and the vanishing of the(higher)p-curvatures,are somewhat different from (and indeed,somewhat surprising from the point of view of the theory of) MF∇-objects.Nevertheless,the properties are of crucial importance in the arguments that underly developments(i),(ii).
新年快乐祝福词§2.The Crystalline Theta Object
2.1.The Complex Analogue.
We begin the discussion of this§by motivating our construction in the abstract algebraic ca byfirst examining the complex analogue of the abstract algebraic theory.
情绪词语Let E be an elliptic curve over C(thefield of complex numbers).In this discussion of the complex analogue,we shall regard E as a complex manifold(rather than an algebraic variety).Let us write O E(respec-tively,O E
)
for the sheaf of complex analytic(respectively,real analytic)
R
