Abstract
Fix a prime $l$ and let $M$ be an integer such that $l\not|M$. Let $f\in S_2(\Gamma_1(M l^2))$ be a newform which is supercuspidal at $l$ of a fixed type related to the nebentypus and special at a finite set of primes. Let $\mathbf{T}^\psi$ be the local quaternionic Hecke algebra associated to $f$. The algebra $\mathbf{T}^\psi$ acts on a module $\mathcal M^\psi_f$ coming from the cohomology of a Shimura curve. It follows from the Taylor-Wiles criterion and a recent Savitt's theorem, that $\mathbf{T}^\psi$ 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 $\mathbf{T}^\psi$. If $f$ occurs at minimal level, we prove a result about congruences of ideals and we obtain a raising the level result. The extension of these results to the non minimal case is still an open problem.
Citation
Miriam Ciavarella. "Congruences between modular forms and related modules." Funct. Approx. Comment. Math. 41 (1) 55 - 70, September 2009. https://doi.org/10.7169/facm/1254330159
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