Cohomology of groups acting on vector spaces over finite fields

TL;DR

This paper provides an effective proof of Nori's theorem on the vanishing of first cohomology groups for groups acting on vector spaces over finite fields, establishing the optimal constant and extending the result.

Contribution

It offers the first effective proof of Nori's theorem with the optimal constant and generalizes the result to all powers of the prime, improving understanding of group actions on finite field vector spaces.

Findings

Proves the optimal constant c(n)=n+2 for Nori's theorem.

Extends the theorem to all powers q of p under the same conditions.

Refines criteria for abelian varieties over number 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.