Congruences between modular forms and related modules

Ciavarella, Miriam

Bollettino della Unione matematica italiana, Série 8, 9B (2006) no. 2, pp. 507-514

Résumé

We fix $\ell$ a prime and let $M$ be an integer such that $\ell \operatorname{\not|} M$; let $f \in S_2(\Gamma_1(M\ell^2))$ be a newform supercuspidal of fixed type at $\ell$ and special at a finite set of primes. For an indefinite quaternion algebra over $Q$, of discriminant dividing the level of $f$, there is a local quaternionic Hecke algebra $T$ associated to $f$. The algebra $T$ acts on a module $M_f$ coming from the cohomology of a Shimura curve. Applying the Taylor-Wiles criterion and a recent Savitt's theorem, $T$ is the universal deformation ring of a global Galois deformation problem associated to $\bar\rho_f$. Moreover $M_f$ is free of rank 2 over $T$. If $f$ occurs at minimal level, as a consequence of our results 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.