CM values of higher automorphic Green functions for orthogonal groups

TL;DR

This paper proves a partial average version of Gross and Zagier's conjecture on CM values of automorphic Green functions for orthogonal groups, extending to Shimura varieties and higher Heegner divisors.

Contribution

It establishes a partial average formula for CM values of automorphic Green functions on orthogonal groups and applies this to higher Heegner divisors on Kuga-Sato varieties.

Findings

Proved a partial average version of Gross-Zagier conjecture for automorphic Green functions.

Extended results to automorphic Green functions on Shimura varieties associated with GSpin(n,2).

Established a Gross-Kohnen-Zagier theorem for higher Heegner divisors.

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 $\mathrm{GSpin}(n,2)$. We also use our approach to prove a Gross-Kohnen-Zagier theorem for higher Heegner divisors on Kuga-Sato varieties over modular curves.