In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense (it roughly means that the action on the tangent space at the identity element of A is a direct sum of onedimensional modules). David Hilbert is said to have remarked the theory constitutes the "most beautiful part of mathematics". Main article: History of mathematics The evolution of mathematics can be seen to be an ever increasing series of abstractions. ...
In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is nonsingular; that is, its graph has no cusps or selfintersections. ...
In abstract algebra, one associates to certain objects a ring, the objects endomorphism ring, which encodes several internal properties of the object. ...
The integers consist of the positive natural numbers (1, 2, 3, â€¦), their negatives (âˆ’1, âˆ’2, âˆ’3, ...) and the number zero. ...
In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ...
The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ...
In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...
In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...
In abstract algebra, a module is a generalization of a vector space. ...
David Hilbert David Hilbert (January 23, 1862 â€“ February 14, 1943) was a German mathematician born in Wehlau, near KÃ¶nigsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ...
Indeed, it is no accident that or equivalently, is so close to an integer. This remarkable fact is explained by the theory of complex multiplication, together with some knowledge of modular forms, and the fact that A modular form is an analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition. ...
is a unique factorization domain. In mathematics, a unique factorization domain (UFD) is, roughly speaking, a commutative ring in which every element can be uniquely written as a product of prime elements, analogous to the fundamental theorem of arithmetic for the integers. ...
An example of an elliptic curve with complex multiplication is  C/Z[i]θ
where Z[i] is the Gaussian integer ring, and θ is any nonzero complex number. Any such complex torus has the Gaussian integers as endomorphism ring. It is known that the corresponding curves can all be written as A Gaussian integer is a complex number whose real and imaginary part are both integers. ...
// Geometry In geometry, a torus (pl. ...
 Y^{2} = 4X^{3} − aX,
having an order 4 automorphism sending In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...
 Y → −iY, X → −X
in line with the action of i on the Weierstrass elliptic functions. This is a typical elliptic curve with complex multiplication, in the sense that over the complex number field they are all found as such quotients, in which some order in the ring of integers in an imaginary quadratic field takes the place of the Gaussian integers. In mathematics, Weierstrass introduced some particular elliptic functions that have become the basis for the most standard notations used. ...
In mathematics, an order in the sense of ring theory in a ring R that is a finitedimensional algebra over the rational number field Q is a subring O of R that satisfies the conditions O spans R over Q, so that QO = R; and O is a lattice...
In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form a0xn + a1xn−1 + ··· + an −1x + an = 0 where n is a positive integer called the degree...
In mathematics, a quadratic field is a field extension K/Q of the form where d is a nonzero rational number. ...
When the base field is a finite field, there are always nontrivial endomorphisms of an elliptic curve; so the complex multiplication case is in a sense typical (and the terminology isn't often applied). But when the base field is a number field, complex multiplication is the exception. It is known that, in a general sense, the case of complex multiplication is the hardest to resolve for the Hodge conjecture. In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ...
In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days...
The Hodge conjecture is a major unsolved problem of algebraic geometry. ...
Kronecker first postulated that the values of elliptic functions at torsion points should be enough to generate all abelian extensions for imaginary quadratic fields, an idea that went back to Eisenstein in some cases, and even to Gauss. This became known as the Kronecker Jugendtraum; and was certainly what had prompted Hilbert's remark above, since it makes explicit class field theory in the way the roots of unity do for abelian extensions of the rational number field. Many generalisations have been sought of Kronecker's ideas; they do however lie somewhat obliquely to the main thrust of the Langlands philosophy, and there is no definitive statement currently known. Leopold Kronecker Leopold Kronecker (December 7, 1823  December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the natural numbers; all else is the work of man (Bell 1986, p. ...
In complex analysis, an elliptic function is, roughly speaking , a function defined on the complex plane which is periodic in two directions. ...
In abstract algebra, an abelian extension is a field extension for which the associated Galois group is abelian. ...
Ferdinand Gotthold Max Eisenstein (April 16, 1823  October 11, 1852) was a German mathematician. ...
Johann Carl Friedrich Gauss Johann Carl Friedrich Gauss (GauÃŸ) (April 30, 1777 â€“ February 23, 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ...
Hilberts twelfth problem, of the 23 Hilberts problems, is the extension of Kroneckers theorem on abelian extensions of the rational numbers, to any base number field. ...
Class field theory is a branch of algebraic number theory, including most of the major results that were proved in the period about 19001950. ...
In mathematics, the nth roots of unity or de Moivre numbers, named after Abraham de Moivre (1667  1754), are complex numbers located on the unit circle. ...
In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
In mathematics, the Langlands program is a web of farreaching and influential conjectures that connect number theory and the representation theory of certain groups. ...
See also: abelian variety of CMtype, LubinTate formal group, Drinfel'd shtuka. In mathematics, an abelian variety A defined over a field K is said to have CM_type if it has a large enough commutative subring in its endomorphism ring End(A). ...
