Vol. 14, No. 8, 2020

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Supersingular locus of Hilbert modular varieties, arithmetic level raising and Selmer groups

Yifeng Liu and Yichao Tian

Vol. 14 (2020), No. 8, 2059–2119
DOI: 10.2140/ant.2020.14.2059
Abstract

This article has three goals: First, we generalize the result of Deuring and Serre on the characterization of supersingular locus to all Shimura varieties given by totally indefinite quaternion algebras over totally real number fields. Second, we generalize the result of Ribet on arithmetic level raising to such Shimura varieties in the inert case. Third, as an application to number theory, we use the previous results to study the Selmer group of certain triple product motive of an elliptic curve, in the context of the Bloch–Kato conjecture.

Keywords
Hilbert modular varieties, supersingular locus, automorphic forms, level raising, Selmer groups
Mathematical Subject Classification 2010
Primary: 14G35
Secondary: 11G05, 11R34
Milestones
Received: 22 October 2018
Revised: 30 September 2019
Accepted: 26 March 2020
Published: 18 September 2020
Authors
Yifeng Liu
Department of Mathematics
Yale University
New Haven, CT
United States
Yichao Tian
UFR de mathématique et de l’informatique
IRMA, University of Strasbourg
France