{
  "id": "2001.07514",
  "version": "v1",
  "published": "2020-01-21T13:30:17.000Z",
  "updated": "2020-01-21T13:30:17.000Z",
  "title": "On algebraic curves with many automorphisms in characteristic p",
  "authors": [
    "Maria Montanucci"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $\\mathcal{X}$ be an irreducible, non-singular, algebraic curve defined over a field of odd characteristic $p$. Let $g$ and $\\gamma$ be the genus and $p$-rank of $\\mathcal{X}$, respectively. The influence of $g$ and $\\gamma$ on the automorphism group $Aut(\\mathcal{X})$ of $\\mathcal{X}$ is well-known in the literature. If $g \\geq 2$ then $Aut(\\mathcal{X})$ is a finite group, and unless $\\mathcal{X}$ is the so-called Hermitian curve, its order is upper bounded by a polynomial in $g$ of degree four (Stichtenoth). In 1978 Henn proposed a refinement of Stichtenoth's bound of cube order in $g$ up to few exceptions, all having $p$-rank zero. In this paper a further refinement of Henn's result is proposed. First, we prove that if an algebraic curve of genus $g \\geq 2$ has more than $336g^2$ automorphisms then its automorphism group has exactly two short orbits, one tame and one non-tame. Then we show that if $|Aut(\\mathcal{X})| \\geq 900g^2$, the quotient curve $\\mathcal{X}/Aut(\\mathcal{X})_P^{(1)}$ where $P$ is contained in the non-tame short orbit is rational, and the stabilizer of 2 points is either a $p$-group or a prime-to-$p$ group, then the $p$-rank of $\\mathcal{X}$ is equal to zero.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-21T13:30:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G20",
      "20B25"
    ],
    "keywords": [
      "algebraic curve",
      "automorphism group",
      "non-tame short orbit",
      "odd characteristic",
      "quotient curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}