{
  "id": "1805.06317",
  "version": "v1",
  "published": "2018-05-15T11:00:29.000Z",
  "updated": "2018-05-15T11:00:29.000Z",
  "title": "An $\\mathbb{F}_{p^2}$-maximal Wiman's sextic and its automorphisms",
  "authors": [
    "Massimo Giulietti",
    "Motoko Kawakita",
    "Stefano Lia",
    "Maria Montanucci"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1603.06706, arXiv:1703.10592",
  "categories": [
    "math.AG"
  ],
  "abstract": "In 1895 Wiman introduced a Riemann surface $\\mathcal{W}$ of genus $6$ over the complex field $\\mathbb{C}$ defined by the homogeneous equation $\\mathcal{W}:X^6+Y^6+Z^6+(X^2+Y^2+Z^2)(X^4+Y^4+Z^4)-12X^2 Y^2 Z^2=0$, and showed that its full automorphism group is isomorphic to the symmetric group $S_5$. The curve $\\mathcal{W}$ was previously studied as a curve defined over a finite field $\\mathbb{F}_{p^2}$ where $p$ is a prime, and necessary and sufficient conditions for its maximality over $\\mathbb{F}_{p^2}$ were obtained. In this paper we first show that the result of Wiman concerning the automorphism group of $\\mathcal{W}$ holds also over an algebraically closed field $\\mathbb{K}$ of positive characteristic $p$, provided that $p \\geq 7$. For $p=2,3$ the polynomial $X^6+Y^6+Z^6+(X^2+Y^2+Z^2)(X^4+Y^4+Z^4)-12X^2 Y^2 Z^2$ is not irreducible over $\\mathbb{K}$, while for $p=5$ the curve $\\mathcal{W}$ is rational and $Aut(\\mathcal{W}) \\cong PGL(2,\\mathbb{K})$. We also show that the $\\mathbb{F}_{19^2}$-maximal Wiman's sextic $\\mathcal{W}$ is not Galois covered by the Hermitian curve $\\mathcal{H}_{19}$ over $\\mathbb{F}_{19^2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-15T11:00:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G20",
      "14H37"
    ],
    "keywords": [
      "maximal wimans sextic",
      "full automorphism group",
      "riemann surface",
      "finite field",
      "hermitian curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}