{
  "id": "0905.0546",
  "version": "v1",
  "published": "2009-05-05T08:30:00.000Z",
  "updated": "2009-05-05T08:30:00.000Z",
  "title": "Genus 3 curves with many involutions and application to maximal curves in characteristic 2",
  "authors": [
    "Enric Nart",
    "Christophe Ritzenthaler"
  ],
  "comment": "18 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let k=F_q be a finite field of characteristic 2. A genus 3 curve C/k has many involutions if the group of k-automorphisms admits a C_2\\times C_2 subgroup H (not containing the hyperelliptic involution if C is hyperelliptic). Then C is an Artin-Schreier cover of the three elliptic curves obtained as the quotient of C by the nontrivial involutions of H, and the Jacobian of C is k-isogenous to the product of these three elliptic curves. In this paper we exhibit explicit models for genus 3 curves with many involutions, and we compute explicit equations for the elliptic quotients. We then characterize when a triple (E_1,E_2,E_3) of elliptic curves admits an Artin-Schreier cover by a genus 3 curve, and we apply this result to the construction of maximal curves. As a consequence, when q is nonsquare and m=\\lfloor 2 sqrt(q) \\rfloor = 1,5,7 mod 8, we obtain that N_q(3)=1+q+3m. We also show that this occurs for an infinite number of values of q nonsquare.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-05-05T08:30:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G20",
      "14H25"
    ],
    "keywords": [
      "maximal curves",
      "characteristic",
      "artin-schreier cover",
      "application",
      "elliptic curves admits"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0905.0546N"
    }
  }
}