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The Hodge-Arakelov Theory of Elliptic Curves: Global Discretization of Local Hodge Theories

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The Hodge-Arakelov Theory of Elliptic Curves: Global Discretization of Local Hodge Theories The Hodge-Arakelov Theory of Elliptic Curves: Global Discretization of Local Hodge Theories by Shinichi Mochizuki September 1999 Table of Contents Introduction §1. Statement of the Main Results §2. Technical Roots: the Work of Mumford and Zhang §3. Conceptual Roots: the Search for a Global Hodge Theory §3.1. From Absolute Differentiation to Comparison Isomorphisms §3.2. A Function-Theoretic Comparison Isomorphism §3.3. The Meaning of Nonlinearity §3.4. Hodge Theory at Finite Resolution §3.5. Relationship to Ordinary Frobenius Liftings and Anabelian Varieties §4. Guide to the Text §5. Future Directions §5.1. Gaussian Poles and the Theta Convolution §5.2. Higher Dimensional Abelian Varieties and Hyperbolic Curves Chapter I: Torsors in Arakelov Theory §0. Introduction §1. Arakelov Theory in Geometric Dimension Zero §2. Definition and First Properties of Torsors §3. Splittings with Bounded Denominators §4. Examples from Geometry Chapter II: The Galois Action on Torsion Points §0. Introduction §1. Some Elementary Group Theory §2. The Height of an Elliptic Curve §3. The Galois Action on the Torsion of a Tate Curve §4. An Effective Estimate of the Image of Galois 1 Chapter III: The Universal Extension of a Log Elliptic Curve §0. Introduction §1. Definition of the Universal Extension §2. Canonical Splitting at Infinity §3. Canonical Splittings in the Complex Case §4. Hodge-Theoretic Interpretation of the Universal Extension §5. Analytic Continuation of the Canonical Splitting §6. Higher Schottky-Weierstrass Zeta Functions §7. Canonical Schottky-Weierstrass Zeta Functions Chapter IV: Theta Groups and Theta Functions §0. Introduction §1. Mumford’s Algebraic Theta Functions §2. Theta Actions and the Schottky Uniformization §3. Twisted Schottky-Weierstrass Zeta Functions §4. Zhang’s Theory of Metrized Line Bundles §5. Theta Groups and Metrized Line Bundles Chapter V: The Evaluation Map §0. Introduction §1. Construction of Certain Metrized Line Bundles §2. The Definition of the Evaluation Map §3. Extension of the Etale-Integral Structure §4. Linear Relations Among Higher Schottky-Weierstrass Zeta Functions §5. The Determinant of the Evaluation Map §6. The Generic Case Chapter VI: The Scheme-Theoretic Comparison Theorem §0. Introduction §1. Definition of a New Integral Structure at Infinity §2. Compatibility with Base-Change §3. The Comparison Isomorphism in Characteristic Zero §4. The Comparison Isomorphism in Mixed Characteristic Chapter VII: The Geometry of Function Spaces: Systems of Orthogonal Functions §0. Introduction §1. The Orthogonalization Problem §2. Review of Legendre and Hermite Polynomials §3. Discrete Tchebycheff Polynomials and the Fundamental Combinatorial Model §4. The K¨ahler Geometry of a Polarized Elliptic Curve §5. The Relationship Between de Rham and Canonical Schottky-Weierstrass Zeta Functions §6. Differential Calculus on the Theta-Weighted Circle 2 Chapter VIII: The Hodge-Arakelov Comparison Theorem §0. Introduction §1. Averages of Metrics §2. The Legendre Model §3. The Truncated Binomial Model §4. The Combinatorics of the Full Binomial Model §5. The Full Binomial Model §6. Relations Among Various Norms and Zeta Functions §7. The Comparison Isomorphism at the Infinite Prime Chapter IX: The Arithmetic Kodaira-Spencer Morphism §0. Introduction §1. The Complex Case: The Classical “Modular Theory” of the Upper Half-Plane §2. The p-adic Case: The Hodge-Tate Decomposition as an Evaluation Map §3. The Global Arithmetic Case: Application of the Hodge-Arakelov Comparison Isomorphism Appendix: Formal Uniformization of Smooth Abelian Group Schemes 3 Introduction §1. Statement of the Main Results The main result of this paper is a Comparison Theorem (cf. Theorem A below) for elliptic curves, which states roughly that: The space of “polynomial functions” of degree (roughly) <don the uni- versal extension of an elliptic curve maps isomorphically via restriction to the space of (set-theoretic) functions on the d-torsion points of the universal extension. This rough statement is essentially precise for (smooth) elliptic curves over fields of char- acteristic zero. For elliptic curves in mixed characteristic and degenerating elliptic curves, this statement may be made precise (i.e., the restriction map becomes an isomorphism) if one modifies the “integral structure” on the space of polynomial functions in an appro- priate fashion. Similarly, in the case of elliptic curves over the complex numbers, one can ask whether or not one obtains an isometry if one puts natural archimedean metrics on the spaces involved. In this paper, we also compute precisely what modification to the integral structure/metrics in all of these cases (i.e., at p-adic and archimedean primes, and for degenerating elliptic curves) is necessary to obtain an isomorphism (or something very close to an isomorphism). In characteristic zero, the universal extension of an elliptic curve may be regarded as the de Rham cohomology of the elliptic curve, with coefficients in the sheaf of invertible functions on the curve. On the other hand, the torsion points of the elliptic curve may be regarded as a portion of the ´etale cohomology of the elliptic curve.Thus,onemay regard this Comparison Theorem as a sort of isomorphism between the de Rham and ´etale cohomologies of the elliptic curve, given by considering functions on each of the respective cohomology spaces. When regarded from this point of view, this Comparison Theorem may be thought of as a sort of discrete or Arakelov-theoretic analogue of the usual comparison theorems between de Rham and ´etale/singular cohomology in the complex and p-adic cases. This analogy with the “classical” local comparison theorems can be made very precise, and is one of the main topics of Chapter IX. Using this point of view, we apply the Comparison Theorem to construct a global/Arakelov-theoretic analogue for elliptic curves over number fields of the Kodaira-Spencer morphism of a family of elliptic curves over a geometric base. This arithmetic Kodaira-Spencer morphism is studied in Chapter IX. 4 Suppose that E is an elliptic curve over a field K of characteristic zero.Letd be a positive integer, and η ∈ E(K) a torsion point of order not dividing d.Write def L = OE(d · [η]) for the line bundle on E corresponding to the divisor of multiplicity d with support at the point η.Write † E → E M ∇ for the universal extension of the elliptic curve, i.e., the moduli space of pairs ( , M ) M ∇ † consisting of a degree zero line bundle on E, together with a connection M .Thus,E is an affine torsor on E under the module ωE of invariant differentials on E.Inparticular, † 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 sense 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 dE ⊆ E of d-torsion † points of E maps isomorphically to the subscheme dE ⊆ E of d-torsion points of E.Then in its simplest form, the main result of this paper states the following: Theorem Asimple. Let E be an elliptic curve over a field K of characteristic zero. Write † E → E for its universal extension. Let d be a positive integer, and η ∈ E(K) a torsion def point whose order does not divide d. Write L = OE(d · [η]). Then the natural map † Γ(E , L)<d →L| † dE † given by restricting sections of L over E whose torsorial degree is <dto the d-torsion † points of E is a bijection between K-vector spaces of dimension d2. The remainder of the main theorem essentially consists of specifying precisely how one must † modify the integral structure of Γ(E , L)<d over more general bases in order to obtain an isomorphism at the finite and infinite primes of a number field, as well as for degenerating elliptic curves. To state the main theorem in its more general form, it is natural to work over a fine noetherian log scheme Slog (cf. [Kato]). Over the base Slog, we consider what we call a log elliptic curve Clog → Slog (cf. Chapter III, Definition 1.1), i.e., the result of pulling back, via some morphism Slog → Mlog Clog → Mlog Mlog ( 1,0)Z,theuniversal log curve ( 1,0)Z. Here, ( 1,0)Z is the log moduli stack 5 of elliptic curves over Z, equipped with its natural log structure defined by the divisor at infinity, and C→(M1,0)Z is the unique (proper) semi-stable curve of genus 1, which we equip with the log structure defined by the divisor in C which is the pull-back of the divisor at infinity of (M1,0)Z. In the general version of the main theorem, we must use line bundles – which play the role of the line bundle L in Theorem Asimple – equipped with a metric as in [Zh] in a neighborhood of the divisor at infinity D ⊆ S. These metrized line bundles “L” live log log on a “Zhang-theoretic version” E∞,S → S∞ of C → S (cf. Chapter IV, §4,5), and are discussed in detail in Chapter V, §1. For smooth elliptic curves, these line bundles are exactly the same as the line bundle “L” of Theorem Asimple. † Next, it turns out that at finite primes the universal extension E → E is not quite † adequate; instead, one must modify it by multiplying the ωE-portion of E by a factor of † § → d (cf. Chapter V, 2). The resulting object E[d] E is an ωE-torsor which coincides with † E over bases on which d is invertible. In a neighborhood of the divisor at infinity, one † must consider a version of E[d] which is compatible with Zhang’s theory of metrized line † bundles.
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