Congruences between modular forms and related modules

Ciavarella, Miriam

Funct. Approx. Comment. Math., Tome 40 (2009) no. 1, p. 55-70 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2009-09-15
Classification:  modular form,  deformation ring,  Hecke algebra,  quaternion algebra,  congruences,  11F80