{
  "id": "math/0207101",
  "version": "v6",
  "published": "2002-07-11T23:54:50.000Z",
  "updated": "2007-01-24T18:13:55.000Z",
  "title": "Improved upper bounds for the number of points on curves over finite fields",
  "authors": [
    "Everett W. Howe",
    "Kristin E. Lauter"
  ],
  "comment": "LaTex, 40 pages. There was a mistake in Section 7 that invalidated the proofs of two of our results. We correct the error in Section 7, and add an appendix with new proofs of the two results",
  "journal": "Ann. Inst. Fourier (Grenoble) 53, fasc. 6 (2003), 1677-1737; Corrigendum, Ann. Inst. Fourier (Grenoble) 57 (2007) 1019-1021",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We give new arguments that improve the known upper bounds on the maximal number N_q(g) of rational points of a curve of genus g over a finite field F_q for a number of pairs (q,g). Given a pair (q,g) and an integer N, we determine the possible zeta functions of genus-g curves over F_q with N points, and then deduce properties of the curves from their zeta functions. In many cases we can show that a genus-g curve over F_q with N points must have a low-degree map to another curve over F_q, and often this is enough to give us a contradiction. In particular, we able to provide eight previously unknown values of N_q(g), namely: N_4(5) = 17, N_4(10) = 27, N_8(9) = 45, N_{16}(4) = 45, N_{128}(4) = 215, N_3(6) = 14, N_9(10) = 54, and N_{27}(4) = 64. Our arguments also allow us to give a non-computer-intensive proof of the recent result of Savitt that there are no genus-4 curves over F_8 having exactly 27 rational points. Furthermore, we show that there is an infinite sequence of q's such that for every g with 0 < g < log_2 q, the difference between the Weil-Serre bound on N_q(g) and the actual value of N_q(g) is at least g/2.",
  "revisions": [
    {
      "version": "v6",
      "updated": "2007-01-24T18:13:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G20",
      "14G05",
      "14G10",
      "14G15"
    ],
    "keywords": [
      "upper bounds",
      "finite field",
      "zeta functions",
      "rational points",
      "genus-g curve"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......7101H"
    }
  }
}