{
  "id": "1411.2151",
  "version": "v1",
  "published": "2014-11-08T19:05:30.000Z",
  "updated": "2014-11-08T19:05:30.000Z",
  "title": "Genus 3 curves whose Jacobians have endomorphisms by $Q(ζ_7 + \\overlineζ_7)$",
  "authors": [
    "J. William Hoffman",
    "Zhibin Liang",
    "Yukiko Sakai",
    "Haohao Wang"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "In this work we consider constructions of genus three curves $X$ such that $\\mathrm{End}(\\mathrm{Jac}(X)) \\otimes Q$ contains the totally real cubic number field $Q(\\zeta _ 7 + \\overline{\\zeta}_7)$. We construct explicit two-dimensional families defined over $Q(s; t)$ whose generic member is a nonhyperelliptic genus 3 curve with this property. The case when X is hyperelliptic was studied by the authors Hoffman and Wang in a previous work. We calculate the zeta function of one of these curves. Conjecturally this zeta function is described by a modular form.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-08T19:05:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "endomorphisms",
      "zeta function",
      "totally real cubic number field",
      "construct explicit two-dimensional families",
      "generic member"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.2151H"
    }
  }
}