Summary: A Huff curve over a field \(K\) is an elliptic curve defined by the equation \(ax(y^2-1)=by(x^2-1)\) where \(a,b\in K\) are such that \(a^2\ne b^2\). In a similar fashion, a general Huff curve over \(K\) is described by the equation \(x(ay^2-1)=y(bx^2-1)\) where \(a,b\in K\) are such that \(ab(a-b)\ne 0\). In this note we express the number of rational points on these curves over a finite field \(\mathbb {F}_q\) of odd characteristic in terms of Gaussian hypergeometric series \[ _2F_1(\lambda ):=_2F_1\left( \begin{matrix} \phi\! &\! \phi \\\! &\!{} \varepsilon \end{matrix}\Big | \lambda \right) \] where \(\phi\) and \(\varepsilon\) are the quadratic and trivial characters over \(\mathbb {F}_q\), respectively. Consequently, we exhibit the number of rational points on the elliptic curves \(y^2=x(x+a)(x+b)\) over \(\mathbb {F}_q\) in terms of \(_2F_1(\lambda )\). This generalizes earlier known formulas for Legendre, Clausen and Edwards curves.
Furthermore, using these expressions we display several transformations of \(_2F_1\). Finally, we present the exact value of \(_2F_1(\lambda )\) for different \(\lambda\)’s over a prime field \(\mathbb {F}_p\) extending previous results of J. Greene [Trans. Am. Math. Soc. 301, 77–101 (1987; Zbl 0629.12017)] and K. Ono [Trans. Am. Math. Soc. 350, No. 3, 1205–1223 (1998; Zbl 0910.11054)].