A Taylor–Wiles System for Quaternionic Hecke Algebras

Abstract

Let [ell ]>3 be a prime. Fix a regular character χ of F[ell ]2× of order ≤ [ell ] − 1, and an integer M prime to [ell ]. Let f∈S2(Γ0(M[ell ]2)) be a newform which is supercuspidal of type χ at [ell ]. 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 [ell ] and semi-stable outside) of the Galois representation ρ¯f over F¯[ell ] 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.