A family of algebraic curves and Appell series over finite fields (Q6579279)

The authors establish a relation between the number of points on the nonsingular projective curve \(C_{a,b,c,d,e}\) over \(\mathbb{Q}\), given by the affine equation\N\(ax^2 + by^2 = c + dx y + ex^2 y^2\), and the finite field Appell series \(F_{4}^{*}\). Specifically, they prove the existence of isogenies between \(C_{a,b,c,d,\frac{ab}{c}}\) and the twisted Edward family of elliptic curves \(E_{\alpha, \frac{16ab\alpha}{d^{2}}}\). In order to prove this result the authors follow the method used by \textit{J. G. Fuselier} [Proc. Am. Math. Soc. 138, No. 1, 109--123 (2010; Zbl 1222.11058)] and \textit{C. Lennon} [Proc. Am. Math. Soc. 139, No. 6, 1931--1938 (2011; Zbl 1281.11104)].