{
  "id": "2006.16857",
  "version": "v1",
  "published": "2020-06-30T14:49:22.000Z",
  "updated": "2020-06-30T14:49:22.000Z",
  "title": "Cohomology of groups acting on vector spaces over finite fields",
  "authors": [
    "Laura Paladino"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-30T14:49:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vector space",
      "finite field",
      "groups acting",
      "cohomology",
      "sufficient conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}