CM values of higher automorphic Green functions for orthogonal groups

作者：Jan Hendrik Bruinier, Stephan Ehlen, Tonghai Yang · 发表于：Inventiones mathematicae · 年份：2021 · DOI：10.1007/s00222-021-01038-0 · 被引用次数：13 · 研究领域：Advanced Algebra and Geometry、Finite Group Theory Research、Analytic Number Theory Research

Abstract Gross and Zagier conjectured that the CM values (of certain Hecke translates) of the automorphic Green function $$G_s(z_1,z_2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 $$z_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>z</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> over all CM points of a fixed discriminant $$d_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>d</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> (twisted by a genus character), and allow in the second variable the evaluation at individual CM points of discriminant $$d_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>d</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> . This result is deduced from more general statements for automorphic Green functions on Shimura varieties associated with the group $${\text {GSpin}}(n,2)$$ <mml:math xmlns:mml="http://www.w3...