Chow groups and L-derivatives of automorphic motives for unitary groups

• 成果类型: Article
• 署名作者: Li, Chao; Liu, Yifeng
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2021.194.3.6
• 发表日期: 2021
• 页码: 817-901
• 关键词: theta-correspondence; eisenstein series; cycles; REPRESENTATIONS; forms

摘要

In this article, we study the Chow group of the motive associated to a tempered global L-packet pi of unitary groups of even rank with respect to a CM extension, whose global root number is -1. We show that, under some restrictions on the ramification of pi, if the central derivative L'(1/2, pi) is nonvanishing, then the pi-nearly isotypic localization of the Chow group of a certain unitary Shimura variety over its reflex field does not vanish. This proves part of the Beilinson-Bloch conjecture for Chow groups and L-functions, which generalizes the Birch and Swinnerton-Dyer conjecture. Moreover, assuming the modularity of Kudla's generating functions of special cycles, we explicitly construct elements in a certain pi-nearly isotypic subspace of the Chow group by arithmetic theta lifting, and compute their heights in terms of the central derivative L'(1/2, pi) and local doubling zeta integrals. This confirms the conjectural arithmetic inner product formula proposed by one of us, which generalizes the Gross-Zagier formula to higher dimensional motives.