Elliptic curve

In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form which is non-singular; that is, the curve has no cusps or self-intersections. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to comprise all non-singular cubic curves; see § Elliptic curves over a general field below.) Formally, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is an abelian variety – that is, it has a multiplication defined algebraically, with respect to which it is an abelian group – and O serves as the identity element. Often the curve itself, without O specified, is called an elliptic curve; the point O is often taken to be the curve's 'point at infinity' in the projective plane. If y2 = P(x), where P is any polynomial of degree three in x with no repeated roots, the solution set is a nonsingular plane curve of genus one, an elliptic curve. If P has degree four and is square-free this equation again describes a plane curve of genus one; however, it has no natural choice of identity element. More generally, any algebraic curve of genus one, for example from the intersection of two quadric surfaces embedded in three-dimensional projective space, is called an elliptic curve, provided that it has at least one rational point to act as the identity. Using the theory of elliptic functions, it can be shown that elliptic curves defined over the complex numbers correspond to embeddings of the torus into the complex projective plane. The torus is also an abelian group, and in fact this correspondence is also a group isomorphism. Elliptic curves are especially important in number theory, and constitute a major area of current research; for example, they were used in the proof, by Andrew Wiles, of Fermat's Last Theorem. They also find applications in elliptic curve cryptography (ECC) and integer factorization. An elliptic curve is not an ellipse: see elliptic integral for the origin of the term. Topologically, a complex elliptic curve is a torus. Although the formal definition of an elliptic curve is fairly technical and requires some background in algebraic geometry, it is possible to describe some features of elliptic curves over the real numbers using only introductory algebra and geometry. In this context, an elliptic curve is a plane curve defined by an equation of the form