Cohomology of groups acting on vector spaces over finite fields

Laura Paladino

Published 2020-06-30Version 1

Let $p$ be a prime and $m,n$ be positive integers. Let $G$ be a group acting on a vector space of dimension $n$ over the finite field ${\mathbf{F}}_q$ with $q=p^m$ elements. A famous theorem proved by Nori in 1987 states that if $m=1$ and $G$ acts semisimply on ${\mathbf{F}}_p^n$, then there exists a constant $c(n)$ depending only on $n$, such that if $p>c(n)$ then $H^1(G,{\mathbf{F}}_p^n)=0$. 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 ${\mathbf{F}}_q^n$ and $p>(2n+1)^2$, then $H^1(G,{\mathbf{F}}_q^n)$ 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.