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

Creators: Liu, Yifeng; Tian, Yichao; Xiao, Liang; Zhang, Wei; Zhu, Xinwen

Abstract

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.

Additional Information

© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022. Received 19 June 2020. Accepted 14 October 2021. Published 21 January 2022. This article is the main outcome of the AIM SQuaREs project Geometry of Shimura varieties and arithmetic application to L-functions conducted by the five authors from 2017 to 2019. We would like to express our sincere gratitude and appreciation to the American Institute of Mathematics for their constant and generous support of the project, and to the staff members at the AIM facility in San Jose, California for their excellent coordination and hospitality. We would like to thank Sug Woo Shin and Yihang Zhu for the discussion concerning Hypothesis 3.2.10 and the endoscopic classification for unitary groups, Zipei Nie and Jun Su for suggesting a proof of a combinatorial lemma (Lemma B.3.3), Ana Caraiani and Peter Scholze for the discussion concerning Sect. D.1, Kai-Wen Lan for the discussion concerning the reference [43], and Ruiqi Bai and Murilo Corato Zanarella for correcting some errors in early drafts. Finally, we thank the anonymous referees for careful reading and many valuable suggestions. The research of Y. L. is partially supported by the NSF Grant DMS-1702019 and a Sloan Research Fellowship. The research of L. X. is partially supported by the NSF Grants DMS-1502147 and DMS-1752703, the Chinese NSF grant under agreement No. NSFC-12071004, and a grant from the Chinese Ministry of Education. The research of W. Z. is partially supported by the NSF Grants DMS-1838118 and DMS-1901642. The research of X. Z. is partially supported by the NSF Grant DMS-1902239 and a Simons Fellowship.

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