Integer Points on Cubic Curves

Abstract

Let C be a non-singular cubic curve given by an equation $$\displaystyle{ax^{3} + bx^{2}y + cxy^{2} + dy^{3} + ex^{2} + fxy + gy^{2} + hx + iy + j = 0}$$ 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.