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

TL;DR

This paper investigates the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives, linking the non-vanishing of central L-values to the structure of Bloch-Kato Selmer groups within the automorphic framework.

Contribution

It establishes new connections between the non-vanishing of Rankin-Selberg L-values and the structure of associated Selmer groups, advancing the understanding of the conjecture in automorphic contexts.

Findings

Non-vanishing of L-values implies vanishing of Selmer groups.

Non-vanishing of certain cycle classes suggests Selmer groups are of rank one.

Results are conditional on conjectural equivalences related to derivatives of L-functions.

Abstract

In this article, we study the Beilinson-Bloch-Kato conjecture for motives corresponding to the Rankin-Selberg product 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 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.