{
  "id": "1312.5108",
  "version": "v1",
  "published": "2013-12-18T12:16:08.000Z",
  "updated": "2013-12-18T12:16:08.000Z",
  "title": "Large $3$-groups of automorphisms of algebraic curves in characteristic $3$",
  "authors": [
    "Massimo Giulietti",
    "Gabor Korchmaros"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $S$ be a $p$-subgroup of the $\\K$-automorphism group $\\aut(\\cX)$ of an algebraic curve $\\cX$ of genus $\\gg\\ge 2$ and $p$-rank $\\gamma$ defined over an algebraically closed field $\\mathbb{K}$ of characteristic $p\\geq 3$.In this paper we prove that if $|S|>2(\\gg-1)$ then one of the following cases occurs. \\begin{itemize} \\item[(i)] $\\gamma=0$ and the extension $\\K(\\cX)/\\K(\\cX)^S$ completely ramifies at a unique place, and does not ramify elsewhere. \\item[(ii)] $\\gamma>0$, $p=3$, $\\cX$ is a general curve, $S$ attains the Nakajima's upper bound $3(\\gamma-1)$ and $\\K(\\cX)$ is an unramified Galois extension of the function field of a general curve of genus $2$ with equation $Y^2=cX^6+X^4+X^2+1$ where $c\\in\\K^*$. \\end{itemize} Case (i) was investigated by Stichtenoth, Lehr, Matignon, and Rocher.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-18T12:16:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H37"
    ],
    "keywords": [
      "algebraic curve",
      "characteristic",
      "general curve",
      "nakajimas upper bound",
      "function field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.5108G"
    }
  }
}