{
  "id": "1702.08812",
  "version": "v1",
  "published": "2017-02-28T15:06:21.000Z",
  "updated": "2017-02-28T15:06:21.000Z",
  "title": "Algebraic curves with many automorphisms",
  "authors": [
    "Massimo Giulietti",
    "Gabor Korchmaros"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $X$ be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus $g \\ge 2$ defined over an algebraically closed field $K$ of odd characteristic $p$. Let $Aut(X)$ be the group of all automorphisms of $X$ which fix $K$ element-wise. It is known that if $|Aut(X)|\\geq 8g^3$ then the $p$-rank (equivalently, the Hasse-Witt invariant) of $X$ is zero. This raises the problem of determining the (minimum-value) function $f(g)$ such that whenever $|Aut(X)|\\geq f(g)$ then $X$ has zero $p$-rank. For {\\em{even}} $g$ we prove that $f(g)\\leq 900 g^2$. The {\\em{odd}} genus case appears to be much more difficult although, for any genus $g\\geq 2$, if $Aut(X)$ has a solvable subgroup $G$ such that $|G|>126 g^2$ then $X$ has zero $p$-rank and $G$ fixes a point of $X$. Our proofs use the Hurwitz genus formula and the Deuring Shafarevich formula together with a few deep results from finite group theory characterizing finite simple groups whose Sylow $2$-subgroups have a cyclic subgroup of index $2$. We also point out some connections with the Abhyankar conjecture and the Katz-Gabber covers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-28T15:06:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H37"
    ],
    "keywords": [
      "algebraic curve",
      "automorphisms",
      "group theory characterizing finite simple",
      "theory characterizing finite simple groups",
      "finite group theory characterizing finite"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}