{
  "id": "1210.8239",
  "version": "v3",
  "published": "2012-10-31T06:19:46.000Z",
  "updated": "2013-09-26T07:08:28.000Z",
  "title": "Counting points on hyperelliptic curves in average polynomial time",
  "authors": [
    "David Harvey"
  ],
  "comment": "17 pages, some simplifications, main theorem strengthened slightly, to appear in the Annals of Mathematics",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let g >= 1 and let Q be a monic, squarefree polynomial of degree 2g + 1 in Z[x]. For an odd prime p not dividing the discriminant of Q, let Z_p(T) denote the zeta function of the hyperelliptic curve of genus g over the finite field F_p obtained by reducing the coefficients of the equation y^2 = Q(x) modulo p. We present an explicit deterministic algorithm that given as input Q and a positive integer N, computes Z_p(T) simultaneously for all such primes p < N, whose average complexity per prime is polynomial in g, log N, and the number of bits required to represent Q.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-09-26T07:08:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G25",
      "11Y99"
    ],
    "keywords": [
      "average polynomial time",
      "hyperelliptic curve",
      "counting points",
      "explicit deterministic algorithm",
      "degree 2g"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.8239H"
    }
  }
}