{
  "id": "1708.03652",
  "version": "v1",
  "published": "2017-08-11T18:25:30.000Z",
  "updated": "2017-08-11T18:25:30.000Z",
  "title": "Non-ordinary curves with a Prym variety of low $p$-rank",
  "authors": [
    "Turku Ozlum Celik",
    "Yara Elias",
    "Burcin Gunes",
    "Rachel Newton",
    "Ekin Ozman",
    "Rachel Pries",
    "Lara Thomas"
  ],
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "If $\\pi: Y \\to X$ is an unramified double cover of a smooth curve of genus $g$, then the Prym variety $P_\\pi$ is a principally polarized abelian variety of dimension $g-1$. When $X$ is defined over an algebraically closed field $k$ of characteristic $p$, it is not known in general which $p$-ranks can occur for $P_\\pi$ under restrictions on the $p$-rank of $X$. In this paper, when $X$ is a non-hyperelliptic curve of genus $g=3$, we analyze the relationship between the Hasse-Witt matrices of $X$ and $P_\\pi$. As an application, when $p \\equiv 5 \\bmod 6$, we prove that there exists a curve $X$ of genus $3$ and $p$-rank $f=3$ having an unramified double cover $\\pi:Y \\to X$ for which $P_\\pi$ has $p$-rank $0$ (and is thus supersingular); for $3 \\leq p \\leq 19$, we verify the same for each $0 \\leq f \\leq 3$. Using theoretical results about $p$-rank stratifications of moduli spaces, we prove, for small $p$ and arbitrary $g \\geq 3$, that there exists an unramified double cover $\\pi: Y \\to X$ such that both $X$ and $P_\\pi$ have small $p$-rank.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-11T18:25:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G20",
      "14H10",
      "14H40",
      "14K15",
      "14Q05"
    ],
    "keywords": [
      "prym variety",
      "non-ordinary curves",
      "unramified double cover",
      "principally polarized abelian variety",
      "smooth curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}