Congruences between modular forms and related modules

Abstract

We fix $\ell$ a prime and let $M$ be an integer such that $\ell \nmid M$; 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 $\TT^ψ$ be the local quaternionic Hecke algebra associated to $f$. The algebra $\TT^ψ$ acts on a module ${\mathcal{M}}_f^{\psi }$ coming from the cohomology of a Shimura curve. Applying the Taylor-Wiles criterion and a recent Savitt's theorem, $\TT^ψ$ is the universal deformation ring of a global Galois deformation problem associated to $\orho_f$. Moreover ${\mathcal{M}}_f^{\psi }$ is free of rank 2 over $\TT^ψ$. 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.