{
  "id": "1501.03526",
  "version": "v1",
  "published": "2015-01-14T21:54:40.000Z",
  "updated": "2015-01-14T21:54:40.000Z",
  "title": "Edwards Curves and Gaussian Hypergeometric Series",
  "authors": [
    "Mohammad Sadek",
    "Nermine El-Sissi"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $E$ be an elliptic curve described by either an Edwards model or a twisted Edwards model over $\\mathbb{F}_p$, namely, $E$ is defined by one of the following equations $x^2+y^2=a^2(1+x^2y^2),\\, a^5-a\\not\\equiv 0$ mod $p$, or, $ax^2+y^2=1+dx^2y^2,\\,ad(a-d)\\not\\equiv0$ mod $p$, respectively. We express the number of rational points of $E$ over $\\mathbb{F}_p$ using the Gaussian hypergeometric series $\\displaystyle {_2F_1}\\left(\\begin{matrix} \\phi&\\phi {} & \\epsilon \\end{matrix}\\Big| x\\right)$ where $\\epsilon$ and $\\phi$ are the trivial and quadratic characters over $\\mathbb{F}_p$ respectively. This enables us to evaluate $|E(\\mathbb{F}_p)|$ for some elliptic curves $E$, and prove the existence of isogenies between $E$ and Legendre elliptic curves over $\\mathbb{F}_p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-14T21:54:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gaussian hypergeometric series",
      "edwards curves",
      "legendre elliptic curves",
      "rational points",
      "quadratic characters"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}