{
  "id": "1110.4693",
  "version": "v1",
  "published": "2011-10-21T03:11:46.000Z",
  "updated": "2011-10-21T03:11:46.000Z",
  "title": "On the distribution of the number of points on a family of curves over finite fields",
  "authors": [
    "Kit-Ho Mak",
    "Alexandru Zaharescu"
  ],
  "comment": "23 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $p$ be a large prime, $\\ell\\geq 2$ be a positive integer, $m\\geq 2$ be an integer relatively prime to $\\ell$ and $P(x)\\in\\mathbb{F}_p[x]$ be a polynomial which is not a complete $\\ell'$-th power for any $\\ell'$ for which $GCD(\\ell',\\ell)=1$. Let $\\mathcal{C}$ be the curve defined by the equation $y^{\\ell}=P(x)$, and take the points on $\\mathcal{C}$ to lie in the rectangle $[0,p-1]^2$. In this paper, we study the distribution of the number of points on $\\mathcal{C}$ inside a small rectangle among residue classes modulo $m$ when we move the rectangle around in $[0,p-1]^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-10-21T03:11:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G20",
      "11T55"
    ],
    "keywords": [
      "finite fields",
      "distribution",
      "residue classes modulo",
      "integer relatively prime",
      "small rectangle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.4693M"
    }
  }
}