Lifting modp representations to characteristics p2

AbstractThere is an abundance of Galois representations in characteristic p that arise as the modp reduction of a characteristic zero representation from algebraic geometry. Except for two-dimensional representations there is little known about the set of modp representations that should arise this way. As a first step in this direction, we consider the problem of finding lifts to characteristic p2 for a representation ρ̄:GK≔Gal(Ksep/K)→GLn(κ), where κ is a finite field of characteristic p,K a local or global field and n any positive integer.If K is a local field, we can show that such lifts always exist. However if p|n, one cannot always fix the determinant of a lift. We also present some partial results for the existence of lifts to characteristic zero.For global fields K, we can construct lifts only if, vaguely speaking, ‘the prime-to-p image of ρ̄ is large inside GLn(κ)’. A sufficient condition for this is the vanishing of H1(Im(ρ̄′),Mn(κ)), where ρ̄′ is the restriction of ρ̄ to GK(ζp) and the action on Mn(κ) is the adjoint action. Based on methods of Cline, Parshall and Scott, we will give a group theoretic criterion for this first cohomology group to vanish