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

摘要

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 is an element of K are such that a(2) not equal 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 is an element of K are such that ab(a-b) not equal 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 F-2(1)(lambda) := F-2(1)((phi phi)(epsilon)vertical bar lambda) where phi and epsilon 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 Fq in terms of F-2(1)(lambda). This generalizes earlier known formulas for Legendre, Clausen and Edwards curves. Furthermore, using these expressions we display several transformations of F-2(1). Finally, we present the exact value of F-2(1)(lambda) for different lambda's over a prime field F-p extending previous results of Greene and Ono.