paper

CM values of higher automorphic Green functions for orthogonal groups

arXiv:1912.12084

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

Revised and final version, to appear in Invent. Math., 71 pages