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 $x(ay^2-1)=y(bx^2-1)$ 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.