Congruences between modular forms and related modules

We fix $\mathrm{\ell }$ a prime and let $M$ be an integer such that $\mathrm{\ell }\mathrm{\mid ̸}M$; let $f\in S_2({\mathrm{\Gamma }}_1(M{\mathrm{\ell }}^2))$ be a newform supercuspidal of fixed type related to the nebentypus, at $\mathrm{\ell }$ and special at a finite set of primes. Let ${\text{TT}}^{\psi }$ be the local quaternionic Hecke algebra associated to $f$. The algebra ${\text{TT}}^{\psi }$ 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, ${\text{TT}}^{\psi }$ is the universal deformation ring of a global Galois deformation problem associated to ${\text{orho}}_f$. Moreover $\mathcal M^\psi_f$isfreeofrank2over$\TT^\psi$\mathrm{.}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.