{
  "id": "0804.0808",
  "version": "v2",
  "published": "2008-04-04T20:39:04.000Z",
  "updated": "2008-10-07T20:18:45.000Z",
  "title": "The fluctuations in the number of points on a hyperelliptic curve over a finite field",
  "authors": [
    "P. Kurlberg",
    "Z. Rudnick"
  ],
  "comment": "10 pages. Final version",
  "categories": [
    "math.NT"
  ],
  "abstract": "The number of points on a hyperelliptic curve over a field of $q$ elements may be expressed as $q+1+S$ where $S$ is a certain character sum. We study fluctuations of $S$ as the curve varies over a large family of hyperelliptic curves of genus $g$. For fixed genus and growing $q$, Katz and Sarnak showed that $S/\\sqrt{q}$ is distributed as the trace of a random $2g\\times 2g$ unitary symplectic matrix. When the finite field is fixed and the genus grows, we find that the the limiting distribution of $S$ is that of a sum of $q$ independent trinomial random variables taking the values $\\pm 1$ with probabilities $1/2(1+q^{-1})$ and the value 0 with probability $1/(q+1)$. When both the genus and the finite field grow, we find that $S/\\sqrt{q}$ has a standard Gaussian distribution.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-10-07T20:18:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11L40",
      "11G25"
    ],
    "keywords": [
      "hyperelliptic curve",
      "fluctuations",
      "independent trinomial random variables",
      "finite field grow",
      "unitary symplectic matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0804.0808K"
    }
  }
}