Rational Points on Elliptic Curves y²=x³+a³ in f_p where p≡1(mod6) is Prime

Abstract

In this work, we consider the rational points on elliptic curves over finite fields F_p. We give results concerning the number of points on the elliptic curve y²≡x³+a³(mod p)where p is a prime congruent to 1 modulo 6. Also some results are given on the sum of abscissae of these points. We give the number of solutions to y²≡x³+a³(modp), also given in ([1], p.174), this time by means of the quadratic residue character, in a different way, by using the cubic residue character. Using the Weil conjecture, one can generalize the results concerning the number of points in F_p to F_p^r.