On the Beilinson–Bloch–Kato conjecture for Rankin–Selberg motives(Yichao Tian)
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.
Inventiones mathematicae volume 228, pages107–375 (2022)
China Institute for Advanced Study in Mathematics, Zhejiang University, Hangzhou 310058
100190, China Morningside Center of Mathematics, AMSS, Chinese Academy of Sciences, Beijing
100871, China Beijing International Center for Mathematical Research, Peking University, Beijing
02139, USA Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA
Pasadena, CA 91125, USA Division of Physics, Mathematics and Astronomy, California Institute of Technology