{
  "id": "1805.08475",
  "version": "v1",
  "published": "2018-05-22T09:54:02.000Z",
  "updated": "2018-05-22T09:54:02.000Z",
  "title": "Evaluation of Gaussian hypergeometric series using Huff's models of elliptic curves",
  "authors": [
    "Mohammad Sadek",
    "Nermine El-Sissi",
    "Arman Shamsi Zargar",
    "Naser Zamani"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "A Huff curve over a field $K$ is an elliptic curve defined by the equation $ax(y^2-1)=by(x^2-1)$ where $a,b\\in K$ are such that $a^2\\ne b^2$. In a similar fashion, a general Huff curve over $K$ is described by the equation $x(ay^2-1)=y(bx^2-1)$ where $a,b\\in K$ are such that $ab(a-b)\\ne 0$. In this note we express the number of rational points on these curves over a finite field $\\mathbb{F}_q$ of odd characteristic in terms of Gaussian hypergeometric series $\\displaystyle {_2F_1}(\\lambda):={_2F_1}\\left(\\begin{matrix} \\phi&\\phi & \\epsilon \\end{matrix}\\Big| \\lambda \\right)$ where $\\phi$ and $\\epsilon$ are the quadratic and trivial characters over $\\mathbb{F}_q$, respectively. Consequently, we exhibit the number of rational points on the elliptic curves $y^2=x(x+a)(x+b)$ over $\\mathbb{F}_q$ in terms of ${_2F_1}(\\lambda)$. This generalizes earlier known formulas for Legendre, Clausen and Edwards curves. Furthermore, using these expressions we display several transformations of ${_2F_1}$. Finally, we present the exact value of $_2F_1(\\lambda)$ for different $\\lambda$'s over a prime field $\\mathbb{F}_p$ extending previous results of Greene and Ono.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-22T09:54:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gaussian hypergeometric series",
      "elliptic curve",
      "huffs models",
      "rational points",
      "evaluation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}