{
  "id": "math/0403178",
  "version": "v1",
  "published": "2004-03-10T15:01:07.000Z",
  "updated": "2004-03-10T15:01:07.000Z",
  "title": "Pointless curves of genus three and four",
  "authors": [
    "Everett W. Howe",
    "Kristin E. Lauter",
    "Jaap Top"
  ],
  "comment": "LaTeX, 15 pages",
  "journal": "pp. 125-141 in: Algebra, Geometry, and Coding Theory (AGCT 2003) (Y. Aubry and G. Lachaud, eds.), Seminaires et Congres 11, Societe Mathematique de France, Paris, 2005",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "A curve over a field k is pointless if it has no k-rational points. We show that there exist pointless genus-3 hyperelliptic curves over a finite field F_q if and only if q < 26, that there exist pointless smooth plane quartics over F_q if and only if either q < 24 or q = 29 or q = 32, and that there exist pointless genus-4 curves over F_q if and only if q < 50.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-03-10T15:01:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G20",
      "14G05",
      "14G10",
      "14G15"
    ],
    "keywords": [
      "pointless curves",
      "pointless smooth plane quartics",
      "k-rational points",
      "hyperelliptic curves",
      "finite field"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......3178H"
    }
  }
}