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The Ramanujan Journal

, Volume 48, Issue 2, pp 357–368 | Cite as

Evaluation of Gaussian hypergeometric series using Huff’s models of elliptic curves

  • Mohammad SadekEmail author
  • Nermine El-Sissi
  • Arman Shamsi Zargar
  • Naser Zamani
Article
  • 26 Downloads

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.

Keywords

Elliptic curves Huff curves Gaussian hypergeometric functions Rational points 

Mathematics Subject Classification

11D45 11G20 11T24 14H52 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Mohammad Sadek
    • 1
    Email author
  • Nermine El-Sissi
    • 1
  • Arman Shamsi Zargar
    • 2
  • Naser Zamani
    • 2
  1. 1.Department of Mathematics and Actuarial ScienceAmerican University in CairoNew CairoEgypt
  2. 2.Department of Mathematics and Applications, Faculty of ScienceUniversity of Mohaghegh ArdabiliArdabilIran

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