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

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 $x(a{y}^2-1)=y(b{x}^2-1)$ 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 ${}_2{F}_1(\lambda ):={}_2{F}_1\left(\begin{array}{ccc}\varphi & \varphi & \epsilon \end{array}|\lambda \right)$ where $\varphi$ 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 ${}_2{F}_1(\lambda )$. This generalizes earlier known formulas for Legendre, Clausen and Edwards curves. Furthermore, using these expressions we display several transformations of ${}_2{F}_1$. Finally, we present the exact value of ${}_2{F}_1(\lambda )$ for different $\lambda$'s over a prime field ${\mathbb{F}}_p$ extending previous results of Greene and Ono.