On the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives

• 成果类型: Article
• 署名作者: Liu, Yifeng; Tian, Yichao; Xiao, Liang; Zhang, Wei; Zhu, Xinwen
• 署名单位: Zhejiang University; Chinese Academy of Sciences; Peking University; Massachusetts Institute of Technology (MIT); California Institute of Technology
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-021-01088-4
• 发表日期: 2022
• 页码: 107-375
• 关键词: local-global compatibility; Galois representations; trace formula; selmer groups; cycles; functoriality; COHOMOLOGY; VARIETIES; monodromy; PRODUCTS

摘要

In this article, we study the Beilinson-Bloch-Kato conjecture for motives associated to Rankin-Selberg products of conjugate self-dual automorphic representations, within the framework of the Gan-Gross-Prasad conjecture. We show that if the central critical value of the Rankin-Selberg L-function does not vanish, then the Bloch-Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch-Kato Selmer group constructed from a certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanishing of the central critical first derivative of the Rankin-Selberg L-function, then the Bloch-Kato Selmer group is of rank one.