Vol. 16, No. 3, 2022

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Integral period relations and congruences

Jacques Tilouine and Eric Urban

Vol. 16 (2022), No. 3, 647–695

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.

congruences, modular forms, periods
Mathematical Subject Classification 2010
Primary: 11F33, 11F41, 11F75
Received: 24 March 2020
Revised: 14 May 2021
Accepted: 24 June 2021
Published: 9 July 2022
Jacques Tilouine
Universite de Paris XIII (Paris-Nord)
Eric Urban
Department of Mathematics
Columbia University
New York, NY
United States