A Taylor-Wiles System for Quaternionic Hecke Algebras

Abstract

Let ℓ > 3 be a prime. Fix a regular character χ of F ℓ2x of order ≤ ℓ - 1, and an integer M prime to ℓ. Let f ∈ S2 (Γ 0(Mℓ2)) be a newform which is supercuspidal of type χ at ℓ. For an indefinite quaternion algebra over Q of discriminant dividing the level of f, there is a local quaternionic Hecke algebra T of type χ associated to f. The algebra T acts on a quaternionic cohomological module M. We construct a Taylor-Wiles system for M, and prove that T is the universal object for a deformation problem (of type χ at ℓ and semi-stable outside) of the Galois representation ρ̄f over F̄ℓ associated to f; that T is complete intersection and that the module M is free of rank 2 over T. We deduce a relation between the quaternionic congruence ideal of type χ for f and the classical one.