Integer Points on Cubic Curves

Abstract

Let C be a non-singular cubic curve given by an equation $$a{x^3} + b{x^2}y + cx{y^2} + d{y^3} + e{x^2} + fxy + g{y^2} + hx + iy + j = 0$$ with integer coefficients. We have seen that if C has a rational point (possibly at infinity), then the set of all rational points on C forms a finitely generated abelian group. So we can get every rational point on C by starting from some finite set and adding points using the geometrically defined group law.