Rational points on elliptic curves y(2)=x(3)+a(3) in F-P where p equivalent to 1 (mod 6) 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 y(2) equivalent to x(3) + a(3) (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 y(2) equivalent to x(3) + a(3) (mod p), also given in [1, page 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.