CM values of higher automorphic Green functions for orthogonal groups

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...

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Bibliographic Details
Published in:Inventiones mathematicae 2021-09, Vol.225 (3), p.693-785
Main Authors: Bruinier, Jan Hendrik, Ehlen, Stephan, Yang, Tonghai
Format: Article
Language:English
Subjects:
Algebra
Green's functions
Logarithms
Mathematical analysis
Mathematics
Mathematics and Statistics
Variables
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Summary: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.
ISSN:0020-9910
1432-1297
DOI:10.1007/s00222-021-01038-0