Cohomology of groups acting on vector spaces over finite fields

Abstract

Let p be a prime and m, n be positive integers. Let G be a group acting faithfully on a vector space of dimension n over the finite field Fq with q = p m elements. A famous theorem proved by Nori in 1987 states that if m = 1 and G acts semisimply on F n p , then there exists a constant c(n) depending only on n, such that if p > c(n) then H 1 (G, F n p ) = 0. We give an explicit constant c(n) = (2n + 1) 2 and prove a more general version of Nori's theorem, by showing that if G acts semisimply on F n q and p > (2n + 1) 2 , then H 1 (G, F n q ) is trivial, for all q. As a consequence, we give sufficient conditions to have an affirmative answer to a classical question posed by Cassels in the case of abelian varieties over number fields. MSC: 20J06, 11G10of H 1 (G, F n q ), when p = q.