On Ihara’s lemma for Hilbert Modular Varieties

Let ρ be a modulo p representation of the absolute Galois group of a totally real number field. Under the assumptions that ρ has large image and admits a low weight crystalline modular deformation we show that any low weight crystalline deformation of ρ unramified outside a finite set of primes will be modular. We follow the approach of Wiles as generalized by Fujiwara. The main new ingredient is an Ihara type lemma for the local component at ρ of the middle degree cohomology of a Hilbert modular variety. As an application we relate the algebraic p-part of the value at 1 of the adjoint L-function associated to a Hilbert modular newform to the cardinality of the corresponding Selmer group.