Lifting mod p representations to characteristics p2

There is an abundance of Galois representations in characteristic p that arise as the mod p reduction of a characteristic zero representation from algebraic geometry. Except for two-dimensional representations there is little known about the set of mod p 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 , 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 pn, 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 is large inside GLn(κ)’. A sufficient condition for this is the vanishing of , where is the restriction of 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.