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.

Publication:

Inventiones mathematicae volume 228, pages107–375 (2022)

Author:

Yifeng Liu

China Institute for Advanced Study in Mathematics, Zhejiang University, Hangzhou 310058

e-mail: liuyf0719@zju.edu.cn

Yichao Tian

100190, China Morningside Center of Mathematics, AMSS, Chinese Academy of Sciences, Beijing

e-mail: yichaot@math.ac.cn

Liang Xiao

100871, China Beijing International Center for Mathematical Research, Peking University, Beijing

e-mail: lxiao@bicmr.pku.edu.cn

Wei Zhang

02139, USA Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA

e-mail: weizhang@mit.edu

Xinwen Zhu

Pasadena, CA 91125, USA Division of Physics, Mathematics and Astronomy, California Institute of Technology

e-mail: xzhu@caltech.edu