Generalized twisted Edwards curves over finite fields and hypergeometric functions

TL;DR

This paper derives explicit formulas for the number of points on generalized twisted Edwards curves over finite fields, connecting these counts to p-adic hypergeometric functions and elliptic curve Frobenius traces.

Contribution

It introduces new point-count formulas for a broad family of algebraic curves, linking them to hypergeometric functions and elliptic curve properties.

Findings

Expressed point counts in terms of p-adic hypergeometric functions.

Connected point counts to Frobenius traces of elliptic curves.

Provided explicit formulas for all finite field elements.

Abstract

Let $\mathbb{F}_q$ be a finite field with $q$ elements. For $a,b,c,d,e,f \in \mathbb{F}_q^{\times}$, denote by $C_{a,b,c,d,e,f}$ the family of algebraic curves over $\mathbb{F}_q$ given by the affine equation \begin{align*} C_{a,b,c,d,e,f}:ay^2+bx^2+cxy=d+ex^2y^2+fx^3y. \end{align*} The family of generalized twisted Edwards curves is a subfamily of $C_{a,b,c,d,e,f}$. Let $\#C_{a,b,c,d,e,f}(\mathbb{F}_q)$ denote the number of points on $C_{a,b,c,d,e,f}$ over $\mathbb{F}_q$. In this article, we find certain expressions for $\#C_{a,b,c,d,e,f}(\mathbb{F}_q)$ when $af=ce$. If $c^2-4ab\neq 0$, we express $\#C_{a,b,c,d,e,f}(\mathbb{F}_q)$ in terms of a $p$-adic hypergeometric function $\mathbb{G}(x)$ whose values are explicitly known for all $x\in \mathbb{F}_q$. Next, if $c^2-4ab=0$, we express $\#C_{a,b,c,d,e,f}(\mathbb{F}_q)$ in terms of another $p$-adic hypergeometric function and then relate it to the traces of Frobenius endomorphisms of a family of elliptic curves. Furthermore, using the known values of the hypergeometric functions, we deduce some nice formulas for $\#C_{a,b,c,d,e,f}(\mathbb{F}_q)$.