Elliptic Curve

An elliptic curve is a smooth, projective, algebraic curve of genus one defined over a field K, together with a specified point O on the curve, describing points in the Cartesian product of K with itself. When the field's characteristic differs from 2 and 3, the curve can be written as the plane algebraic curve of solutions to y squared equals x cubed plus a times x plus b, for coefficients a and b in K, required to be non-singular, meaning free of cusps or self-intersections. The curve is usually understood as embedded in the projective plane, with O as the unique point at infinity, and it is an abelian variety, meaning it carries a group law defined algebraically under which O serves as the identity element. Using the theory of elliptic functions, elliptic curves defined over the complex numbers can be shown to correspond to embeddings of the torus into the complex projective plane, a correspondence that is also a group isomorphism. Elliptic curves are especially important in number theory and remain a major area of current research; they were used in Andrew Wiles's proof of Fermat's Last Theorem, and they find applications in elliptic curve cryptography and integer factorization. Despite the name, an elliptic curve is not an ellipse in the sense of a projective conic, which has genus zero.