{
  "id": "1306.5102",
  "version": "v1",
  "published": "2013-06-21T11:29:26.000Z",
  "updated": "2013-06-21T11:29:26.000Z",
  "title": "Frobenius lifts and point counting for smooth curves",
  "authors": [
    "Amnon Besser",
    "François-Renaud Escriva",
    "Rob de Jeu"
  ],
  "comment": "29 pages, 2 figures",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "We describe an algorithm to compute the zeta-function of a proper, smooth curve over a finite field, when the curve is given together with some auxiliary data. Our method is based on computing the matrix of the action of a semi-linear Frobenius on the first cohomology group of the curve by means of Serre duality. The cup product involved can be computed locally, after first computing local expansions of a globally defined lift of Frobenius. The resulting algorithm's complexity is softly cubic in the field degree, which is also the case with Kedlaya's algorithm in the hyperelliptic case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-21T11:29:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14F30",
      "14G10",
      "14G15",
      "14Q50",
      "14G22"
    ],
    "keywords": [
      "smooth curve",
      "frobenius lifts",
      "point counting",
      "first computing local expansions",
      "first cohomology group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.5102B"
    }
  }
}