Functiones et Approximatio Commentarii Mathematici

Congruences between modular forms and related modules

Miriam Ciavarella

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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.

Article information

Funct. Approx. Comment. Math., Volume 41, Number 1 (2009), 55-70.

First available in Project Euclid: 30 September 2009

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Primary: 11F80: Galois representations

modular form deformation ring Hecke algebra quaternion algebra congruences


Ciavarella, Miriam. Congruences between modular forms and related modules. Funct. Approx. Comment. Math. 41 (2009), no. 1, 55--70. doi:10.7169/facm/1254330159.

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