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arXiv · 1912.11942

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

Yifeng Liu·Yichao Tian·Liang Xiao·Wei Zhang·Xinwen Zhu
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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.

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Yifeng Liu, Yichao Tian, Liang Xiao, Wei Zhang, Xinwen Zhu. 2019-12-26. On the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives. https://arxiv.org/abs/1912.11942

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