Elliptic Curves

An elliptic curve over the rational numbers is an algebraic curve given by an equation of the form y^2=f(x) where f(x) is a cubic in x with rational coefficients and distinct roots (over the complex numbers). Many Diophantine problems reduce to elliptic curves, for example homogeneous cubics in 3 variables, equations y^2=q(x) for quartic q(x) or intersections of two quadrics in projective 3-space.

A crucial aspect of the theory of elliptic curves is that the points on the curve with coefficients in a fixed field form a group by adding a point O to be the identity and declaring that the sum of three points where a line hits the cubic is always O. This clarifies the chord and tangent process known by Fermat which produces new rational points by intersecting with the line tangent at a rational point or the chord joining two rational points.