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# 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

#### 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