LIFTINGN-DIMENSIONAL GALOIS REPRESENTATIONS TO CHARACTERISTIC ZERO

Abstract

AbstractLetFbe a number field, letN≥ 3 be an integer, and letkbe a finite field of characteristic ℓ. We show that ifρ:GF→GLN(k) is a continuous representation with image ofρcontainingSLN(k) then, under moderate conditions at primes dividing ℓ∞, there is a continuous representation ρ:GF→GLN(W(k)) unramified outside finitely many primes withρ~ρ mod ℓ. Stronger results are presented forρ:Gℚ→GL3(k).