Abstract

Gross and Zagier conjectured that the CM values (of certain Hecke translates) of the automorphic Green function G_s(z_1,z_2) for the elliptic modular group at positive integral spectral parameter s are given by logarithms of algebraic numbers in suitable class fields. We prove a partial average version of this conjecture, where we sum in the first variable z_1 over all CM points of a fixed discriminant d_1 (twisted by a genus character), and allow in the second variable the evaluation at individual CM points of discriminant d_2. This result is deduced from more general statements for automorphic Green functions on Shimura varieties associated with the group {text {GSpin}}(n,2). We also use our approach to prove a Gross–Kohnen–Zagier theorem for higher Heegner divisors on Kuga–Sato varieties over modular curves.

Highlights

  • The automorphic Green function for = SL2(Z), called the resolvent kernel function for, plays an important role in the theory of automorphic forms, see e.g. [21,25]

  • Gs is invariant under the action of in both variables and descends to a function on (X × X ) \ Z (1), where X = \H and Z (1) denotes the diagonal

  • In the present paper we prove stronger results, by only averaging over the CM points z1 of one discriminant d1 and allowing for z2 individual CM points of discriminant d2

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Summary

Introduction

The sum converges absolutely for s ∈ C with (s) > 1, and z1, z2 in the complex upper half-plane H with z1 ∈/ z2. Gs is invariant under the action of in both variables and descends to a function on (X × X ) \ Z (1), where X = \H and Z (1) denotes the diagonal. The differential equation of the Legendre function implies that Gs is an eigenfunction of the hyperbolic Laplacian in both variables. It has a meromorphic continuation in s to the whole complex plane and satisfies a functional equation relating the values at s and 1 − s

The algebraicity conjecture
Higher automorphic Green functions on orthogonal Shimura varietes
A higher weight Gross–Kohnen–Zagier theorem
Orthogonal Shimura varieties and theta functions
Siegel theta functions
Differential operators
Weak Maass forms
Binary theta functions
Automorphic Green functions
Properties of automorphic Green functions
Positive integral values of the spectral parameter
CM values of higher Green functions
General CM cycles
The resolvent kernel
Modular curves
A Gross–Kohnen–Zagier theorem for higher weight Heegner cycles
Partial averages
Twisted special divisors
Twisted Siegel and Millson theta functions
Twisted theta lifts
CM values
Numerical examples
Example 1
Example 2
Example 3
Weakly holomorphic modular forms
Special preimages of binary theta functions
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