Congruences between modular forms and related modules

TL;DR

This paper establishes a deep connection between certain modular forms and Galois deformation rings using the Taylor-Wiles method, and explores level raising and congruence ideals in the context of Shimura curves.

Contribution

It proves that the local quaternionic Hecke algebra is the universal deformation ring for associated Galois representations and extends level raising results to minimal levels.

Findings

The Hecke algebra acts freely of rank 2 on the cohomology module.

The Hecke algebra is isomorphic to the universal deformation ring.

Level raising results are established for minimal levels.

Abstract

We fix $\ell$ a prime and let $M$ be an integer such that $\ell\not|M$; let $f\in S_2(\Gamma_1(M\ell^2))$ be a newform supercuspidal of fixed type related to the nebentypus, at $\ell$ and special at a finite set of primes. Let $\TT^\psi$ be the local quaternionic Hecke algebra associated to $f$. The algebra $\TT^\psi$ acts on a module $\mathcal M^\psi_f$ coming from the cohomology of a Shimura curve. Applying the Taylor-Wiles criterion and a recent Savitt's theorem, $\TT^\psi$ is the universal deformation ring of a global Galois deformation problem associated to $\orho_f$. Moreover $\mathcal M^\psi_f$ is free of rank 2 over $\TT^\psi$. If $f$ occurs at minimal level, by a generalization of a Conrad, Diamond and Taylor's result and by the classical Ihara's lemma, we prove a theorem of raising the level and a result about congruence ideals. The extension of this results to the non minimal case is an open problem.