Positive definite quadratic forms, elliptic curves and cubic congruences

Abstract

Let F(x, y) = ax 2 + bxy + cy 2 be a positive definite binary quadratic form with discriminant Δ whose base points lie on the line x = -1/m for an integer m ≥ 2, let p be a prime number and let F p be a finite field. Let E F: y 2= ax 3 + bx 2 + cx be an elliptic curve over F p and let C F: ax 3 + bx 2 + cx ≡ 0(mod p) be the cubic congruence corresponding to F. In this work we consider some properties of positive definite quadratic forms, elliptic curves and cubic congruences.