Abstract
Let F be a number field. Given a continuous representation ρ ¯ : G F → GL 2 ( F ¯ ℓ ) with insoluble image we show, under moderate assumptions at primes dividing ℓ∞, that ρ ¯ ∼ ρ mod ℓ for some continuous representation ρ : G F → GL 2 ( Q ¯ ℓ ) which is unramified outside finitely many primes. We also establish level lowering when F is totally real, ρ ¯ is the reduction of a nearly ordinary Hilbert modular form and is distinguished at ℓ.
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