>>462
つづき

The main idea here is the following: Assume that we are given an elliptic
curve EK over a number field K, with everywhere semi-stable reduction. Also, let
us assume that all of the d-torsion points of EK are defined over K. The arithmetic
Kodaira-Spencer morphism (cf. §1.4) essentially consists of applying some sort of
Galois action to an Arakelov-theoretic vector bundle HDR on Spec(OK) and seeing
what effect this Galois action has on the natural Hodge filtration Fr(HDR) on HDR.

If one ignores the Gaussian poles, the subquotients (Fr+1/Fr)(HDR) of this Hodge
filtration essentially (“as a function of r”) look like
τ^○xr E
(tensored with some object which is essentially irrelevant since it is independent of
r). Thus, as long as the “arithmetic Kodaira-Spencer is nontrivial” (which it most
surely is!), the Galois action on HDR would give rise to nontrivial globally integral
(in the sense of Arakelov theory) morphisms

P35
Section 2: The Theta Convolution
§2.1. Background
Perhaps the simplest way to explain the main idea of [Mzk2] is the following: The theory of [Mzk1] may be thought of as a sort of discrete, scheme-theoretic
version of the theory of the classical Gaussian e^-x^2
(on the real line) and its derivatives (cf. [Mzk1], Introduction, §2). More concretely, the theory of [Mzk1] may, in
essence, be thought of as the theory of the theta function
Θ def= Σn∈Zq^n^2・ U n
(where q is the q-parameter, and U is the standard multiplicative coordinate on
Gm) and its derivatives ? i.e., functions of the form
Σn∈Z q^n^2・ P(n) ・ U n

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