Elliptic Curve

An elliptic curve is a smooth projective curve of genus one, equivalently the set of solutions to a Weierstrass equation y² = x³ + ax + b over a field. The group law on its points makes it the simplest algebraic variety with a nontrivial abelian group structure. In number theory, elliptic curves are governed by the Birch and Swinnerton-Dyer conjecture, which links the rank of rational points to the L-function of the curve. Andrew Wiles proved Fermat's Last Theorem by establishing a special case of the modularity theorem for elliptic curves.

Elliptic curves are not merely a meeting point of number theory and geometry. They are a test site. Every major conjecture in modern arithmetic has been tested on elliptic curves first. If a theory cannot survive contact with them, it cannot survive at all.