Edwards Curves and Gaussian Hypergeometric Series

TL;DR

This paper expresses the number of rational points on Edwards and twisted Edwards elliptic curves over finite fields using Gaussian hypergeometric series, enabling point counting and establishing isogenies with Legendre elliptic curves.

Contribution

It provides a novel expression for point counts of Edwards curves via hypergeometric series and proves the existence of isogenies with Legendre elliptic curves over finite fields.

Findings

Explicit formulas for |E(𝔽_p)| using hypergeometric series

Evaluation of point counts for specific elliptic curves

Proof of isogenies between Edwards and Legendre elliptic curves

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 $\mathbb{F}_p$.