On higher congruences between automorphic forms

TL;DR

This paper establishes a precise algebraic relation connecting the p-adic valuation of the congruence module's order to p-power congruences among automorphic forms, with applications to Mazur's Eisenstein ideal.

Contribution

It provides a new algebraic framework linking congruence modules to p-adic valuations and automorphic form congruences, extending understanding of automorphic congruences modulo prime powers.

Findings

Exact relation between p-adic valuation and automorphic form congruences

Application to Mazur's Eisenstein ideal

Enhanced understanding of congruences modulo prime powers

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.