{'fraction_non_alphanumeric': 0.0437246963562753, 'fraction_numerical': 0.0008097165991902834, 'mean_word_length': 4.469026548672566, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Lifting mod p representations to characteristics p 2

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 p for a representation ρ̄ : GK := Gal(K/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 H(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.