Generalized twisted Edwards curves over finite fields and hypergeometric functions

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 $c^2-4ab\ne 0$, 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 $\#C_{a,b,c,d,e,f}({\mathbb{F}}_q)$.

27 pages