Evaluation of Gaussian hypergeometric series using Huff’s models of elliptic curves

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∈ K are such that a ² ≠ 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∈ K are such that ab(a- b) ≠ 0. In this note we express the number of rational points on these curves over a finite field F _q of odd characteristic in terms of Gaussian hypergeometric series 2F1(λ):=2F1(ϕϕϵ|λ) where ϕ and ϵ are the quadratic and trivial characters over F _q , respectively. Consequently, we exhibit the number of rational points on the elliptic curves y ² = x(x+ a) (x+ b) over F _q in terms of ₂ F ₁ (λ). This generalizes earlier known formulas for Legendre, Clausen and Edwards curves. Furthermore, using these expressions we display several transformations of ₂ F ₁ . Finally, we present the exact value of ₂ F ₁ (λ) for different λ’s over a prime field F _p extending previous results of Greene and Ono.