Cohomology of groups acting on vector spaces over finite fields

Abstract

Let ${\mathbf{F}}_q$ be the finite field with $q=p^m$ elements and $G$ be a subgroup of ${\rm{GL}}_n({\mathbf{F}}_q)$. A famous theorem of Nori published in 1987 states that there exists a (non-effective) constant $c(n)$, depending only on $n$, such that if $p>c(n)$ and $G$ acts semisimply on ${\mathbf{F}}_p^n$, then $H^1(G,{\mathbf{F}}_p^n)=0$. We solve the long-standing problem, also considered by Serre of giving an effective proof of Nori's Theorem. Our approach yields the optimal constant $c(n)=n+2$. We also prove a more general version of Nori's theorem, namely, that for all powers $q$ of $p$, if $G$ acts semisimply on ${\mathbf{F}}_q^n$ and $p>n+2$, then $H^1(G,{\mathbf{F}}_q^n)$ is trivial. We apply these results to refine a criterion, proved by \c{C}iperiani and Stix, which gives sufficient conditions for an affirmative answer to a classical question posed by Cassels in the case of abelian varieties over number fields.