CM values of higher automorphic Green functions for orthogonal groups

• 成果类型: Article
• 署名作者: Bruinier, Jan Hendrik; Ehlen, Stephan; Yang, Tonghai
• 署名单位: Technical University of Darmstadt; University of Cologne; University of Wisconsin System; University of Wisconsin Madison
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-021-01038-0
• 发表日期: 2021
• 页码: 693-785
• 关键词: kohnen-zagier theorem; shimura varieties; heegner points; maass forms; cycles; derivatives; INTEGRALS; heights; traces

摘要

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.