Congruences between modular forms and related modules

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.