A family of algebraic curves and Appell series over finite fields

For \(a,b,c,d,e\in {\mathbb {Q}}^\times \) $a,b,c,d,e\in {\mathbb{Q}}^{\times }$ , denote by \(C_{a,b,c,d,e}\) $C_{a,b,c,d,e}$ the nonsingular projective curve over \({\mathbb {Q}}\) $\mathbb{Q}$ given by the affine equation \(\begin{aligned} C_{a,b,c,d,e}:\;ax^2+by^2=c+\textrm{d}xy+ex^2y^2. \end{aligned}\) $$\begin{array}{c}\hfill C_{a,b,c,d,e}:a{x}^2+b{y}^2=c+\text{d}xy+e{x}^2{y}^2.\end{array}$$ In this paper, we find a relation between the number of points on \(C_{a,b,c,d,e}\) $C_{a,b,c,d,e}$ over \({\mathbb {F}}_q\) ${\mathbb{F}}_q$ and the finite field Appell series \(F_4^*\) $F_4^{\ast }$ . As a consequence, in view of certain transformations for finite field Appell series \(F_4^*\) $F_4^{\ast }$ , we prove the existence of isogenies between \(C_{a,b,c,d,\frac{ab}{c}}\) $C_{a,b,c,d,\frac{\mathit{ab}}{c}}$ and the twisted Edward family of elliptic curves \(E_{\alpha ,\frac{16ab\alpha }{d^2}}\) $E_{\alpha ,\frac{16ab\alpha }{d^2}}$ .