Rational Points on Elliptic Curves y^2=x^3+a^3 in f_{p} where p{\equiv}1(mod6) is Prime

TL;DR

This paper investigates the number of rational points on specific elliptic curves over finite fields where the prime p is congruent to 1 modulo 6, using quadratic and cubic residue characters, and generalizes results via the Weil conjecture.

Contribution

It provides new methods to count points on elliptic curves over finite fields with primes p ≡ 1 mod 6, employing cubic residue characters and generalizing with Weil conjecture.

Findings

Number of solutions to y^2 ≡ x^3 + a^3 mod p determined

Use of cubic residue character for point counting

Generalization to F_{p^r} via Weil conjecture

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{\equiv}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{\equiv}x^3+a^3(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}}.