Edwards Curves and Gaussian Hypergeometric Series

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 $\mathbb{F}_p$.