ON THE NUMBER OF POINTS ON y^2 = x^3 - ax OVER ð”½_q

Abstract

In [8, Chap. 18, Thm. 5], Rosen and Ireland express the number of ð”½_p points on the family of elliptic curves y² = x³ - ax in terms of Jacobi sums using properties of character sums. In this paper we give an alternative proof of this result using Gaussian hypergeometric series and extend it to ð”½_q. Further if a is a quadratic residue in ð”½_q, then we find a similar results using another technique.