Lifting Galois representations of number fields

Let F be a number field. Given a continuous representation (rho) over bar : G(F) -> GL(2)((F) over bar (l)) with insoluble image we show, under moderate assumptions at primes dividing l(infinity), that (rho) over bar similar to rho mod l for some continuous representation rho : G(F) -> GL(2)((Q) over bar (l)) which is unramified outside finitely many primes. We also establish level lowering when F is totally real, (rho) over bar is the reduction of a nearly ordinary Hilbert modular form and is distinguished at l. (C) 2008 Elsevier Inc. All rights reserved.