{
  "id": "1507.00914",
  "version": "v1",
  "published": "2015-07-03T13:54:14.000Z",
  "updated": "2015-07-03T13:54:14.000Z",
  "title": "Character Sums, Gaussian Hypergeometric Series, and Hyperelliptic Curves",
  "authors": [
    "Mohammad Sadek"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the character sums \\[\\phi_{(m,n)}(a,b)=\\sum_{x\\in\\mathbb{F}_q}\\phi\\left(x(x^{m}+a)(x^{n}+b)\\right),\\textrm{ and, } \\psi_{(m,n)}(a,b)=\\sum_{x\\in\\mathbb{F}_q}\\phi\\left((x^{m}+a)(x^{n}+b)\\right)\\] where $\\phi$ is the quadratic character defined over $\\mathbb{F}_q$. These sums are expressed in terms of Gaussian hypergeometric series over $\\mathbb{F}_q$. Then we use these expressions to exhibit the number of $\\mathbb{F}_q$-rational points on families of hyperelliptic curves and their Jacobian varieties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-03T13:54:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gaussian hypergeometric series",
      "hyperelliptic curves",
      "character sums",
      "quadratic character",
      "rational points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150700914S"
    }
  }
}