Mathematical Research Letters

Volume 21 (2014)

Number 1

On higher congruences between automorphic forms

Pages: 71 – 82

DOI: http://dx.doi.org/10.4310/MRL.2014.v21.n1.a5

Authors

Tobias Berger (School of Mathematics and Statistics, University of Sheffield, United Kingdom)

Krzysztof Klosin (Department of Mathematics, Queens College, City University of New York, Queens, N.Y., U.S.A.)

Kenneth Kramer (Department of Mathematics, Queens College, City University of New York, Queens, N.Y., U.S.A.)

Abstract

We prove a commutative algebra result which has consequences for congruences between automorphic forms modulo prime powers. If $C$ denotes the congruence module for a fixed automorphic Hecke eigenform $\pi_0$, we prove an exact relation between the $p$-adic valuation of the order of $C$ and the sum of the exponents of $p$-power congruences between the Hecke eigenvalues of $\pi_0$ and other automorphic forms. We apply this result to several situations including the congruences described by Mazur’s Eisenstein ideal.

Keywords

congruences, automorphic forms

2010 Mathematics Subject Classification

11F33

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