Beilinson--Bloch--Kato conjecture for some Rankin-Selberg motives.

Abstract: The Birch and Swinnerton-Dyer conjecture is known in the case of rank $0$ and $1$ thanks to the foundational work of Kolyvagin and Gross-Zagier. In this talk, I will report on a joint work with Yifeng Liu, Yichao Tian, Wei Zhang, and Xinwen Zhu. We study the analogue and generalizations of Kolyvagin's result to the unitary Gan-Gross-Prasad paradigm. More precisely, our ultimate goal is to show that, under some technical conditions, if the central value of the Rankin-Selberg $L$-function of an automorphic representation of $U(n) \ast U(n+1)$ is nonzero, then the associated Selmer group is trivial; Analogously, if the Selmer class of certain cycle for the $U(n) \ast U(n+1)$-Shimura variety is nontrivial, then the dimension of the corresponding Selmer group is one. ($\href{https://drive.google.com/file/d/1crDh5JwaFv8HW7wZ3NZtDugaFQHOZrTG/view?usp=sharing}{{\rm notes}}$).

algebraic geometrynumber theoryrepresentation theory

Audience: researchers in the discipline

( slides )

VIASM Arithmetic Geometry Online Seminar