@MISC{Kudla03modularforms, author = {Stephen S. Kudla}, title = {Modular forms and arithmetic geometry}, year = {2003} }

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Abstract

The aim of these notes is to describe some examples of modular forms whose Fourier coefficients involve quantities from arithmetical algebraic geometry. At the moment, no general theory of such forms exists, but the examples suggest that they should be viewed as a kind of arithmetic analogue of theta series and that there should be an arithmetic Siegel–Weil formula relating suitable averages of them to special values of derivatives of Eisenstein series. We will concentrate on the case for which the most complete picture is available, the case of generating series for cycles on the arithmetic surfaces associated to Shimura curves over Q, expanding on the treatment in [40]. A more speculative overview can be found in [41]. In section 1, we review the basic facts about the arithmetic surface M associated to a Shimura curve over Q. These arithmetic surfaces are moduli stacks over Spec(Z) of pairs (A, ι) over a base S, where A is an abelian scheme of relative dimension 2 and ι is an action on A of a maximal order OB in an indefinite quaternion algebra B over Q. In section 2, we recall the definition of the arithmetic Chow group ĈH 1 (M), following Bost, [7], and we discuss the metrized Hodge line bundle ˆω and the conjectural value of 〈ˆω, ˆω〉, where 〈 , 〉 is the height pairing on ĈH1(M). In the next two sections, we describe divisors Z(t), t ∈ Z>0, on M. These are defined as the locus of (A, ι, x)’s where x is a special endomorphism (Definition 3.1) of (A, ι) with x2 = −t. Since such an x gives an action on (A, ι) of the order Z [ √ −t] in the imaginary quadratic fieldkt = Q ( √ −t), the cycles Z(t) can be viewed as analogues of the familiar CM points on modular curves. In section 3, the complex points and hence the horizontal components of Z(t) are determined. In section 4, the vertical components of Z(t) are determined using the p-adic uniformization of the fibers Mp of bad reduction of M. In section 5, we construct Green functions Ξ(t, v) for the divisors Z(t), depending on a parameter v ∈ R ×>0. When t < 0, the series defining Ξ(t, v) becomes a smooth function on M(C). These Green functions are used in section 6 to define classes ̂ Z(t, v) ∈ ĈH1(M), for t ∈ Z, t ̸ = 0, and an additional class ̂ Z(0, v) is defined using ˆω. The main result of section 6 (Theorem 6.3) says