{
  "id": "math/0607308",
  "version": "v3",
  "published": "2006-07-13T13:18:56.000Z",
  "updated": "2007-01-08T16:13:57.000Z",
  "title": "Computing Zeta Functions of Nondegenerate Curves",
  "authors": [
    "Wouter Castryck",
    "Jan Denef",
    "Frederik Vercauteren"
  ],
  "comment": "41 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "In this paper we present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since all known cases, e.g. hyperelliptic, superelliptic and C_{ab} curves, can be transformed to fit the nondegenerate case. For curves with a fixed Newton polytope, the property of being nondegenerate is generic, so that the algorithm works for almost all curves with given Newton polytope. For a genus g curve over F_{p^n}, the expected running time is O(n^3g^6 + n^2g^{6.5}), whereas the space complexity amounts to O(n^3g^4), assuming p is fixed.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2007-01-08T16:13:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "computing zeta functions",
      "nondegenerate curve",
      "space complexity amounts",
      "paper vastly generalizes",
      "finite field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......7308C"
    }
  }
}