A Taylor-Wiles system for quaternionic Hecke algebras

Let l> 3 be a prime. Let f be a newform which is supercuspidal of a fixed type w 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 w 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 w at l and semi-stable outside) of the Galois representation 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 con- gruence ideal of type w for f and the classical one.