Integral period relations and congruences

Tilouine, Jacques; Urban, Eric

doi:10.2140/ant.2022.16.647

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Integral period relations and congruences

Jacques Tilouine and Eric Urban

Vol. 16 (2022), No. 3, 647–695

DOI: 10.2140/ant.2022.16.647

Abstract

Under relatively mild and natural conditions, we establish integral period relations for the (real or imaginary) quadratic base change of an elliptic cusp form. This answers a conjecture of Hida regarding the congruence ideal controlling the congruences between this base change and other eigenforms which are not base change. As a corollary, we establish the Bloch–Kato conjecture for adjoint modular Galois representations twisted by an even quadratic character. In the odd case, we formulate a conjecture linking the degree two topological period attached to the base change Bianchi modular form, the cotangent complex of the corresponding Hecke algebra and the archimedean regulator attached to some Beilinson–Flach element.

Keywords

congruences, modular forms, periods

Mathematical Subject Classification 2010

Primary: 11F33, 11F41, 11F75

Milestones

Received: 24 March 2020

Revised: 14 May 2021

Accepted: 24 June 2021

Published: 9 July 2022

Authors

Jacques Tilouine
LAGA Universite de Paris XIII (Paris-Nord) Villetaneuse France

Eric Urban
Department of Mathematics Columbia University New York, NY United States

Vol. 16, No. 3, 2022