{
  "id": "0803.3534",
  "version": "v2",
  "published": "2008-03-25T12:04:20.000Z",
  "updated": "2008-05-13T11:54:08.000Z",
  "title": "Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field",
  "authors": [
    "Dmitry Faifman",
    "Zeev Rudnick"
  ],
  "comment": "Added references to the CLT in RMT",
  "doi": "10.1112/S0010437X09004308",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann Hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genus g a fixed interval I will contain 2g|I| angles as the genus grows. We show that for the variance of number of angles in I is asymptotically a constant multiple of log(2g|I|) and prove a central limit theorem: The normalized fluctuations are Gaussian. These results continue to hold for shrinking intervals as long as the expected number of angles 2g|I| tends to infinity.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-05-13T11:54:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G20",
      "14G10",
      "15A52"
    ],
    "keywords": [
      "zeta functions",
      "hyperelliptic curves",
      "statistics",
      "central limit theorem",
      "genus grows"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0803.3534F"
    }
  }
}