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In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice. There is also the higher-dimensional complex multiplication theory of abelian varieties A having enough endomorphisms in a certain precise sense, roughly that the action on the tangent space at the identity element of A is a direct sum of one-dimensional modules.

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rdfs:label	Complex multiplication (en) Multiplicación compleja (es) Moltiplicazione complessa (it) Multiplication complexe (fr) 복소 곱셈 (ko) 虚数乗法 (ja) Complexe vermenigvuldiging (nl) Multiplicação complexa (pt)
rdfs:comment	En mathématiques, une courbe elliptique est à multiplication complexe si l'anneau de ses endomorphismes est plus grand que celui des entiers (il existe une théorie plus générale de la multiplication complexe pour les variétés abéliennes de dimension supérieure). Cette notion est liée au douzième problème de Hilbert. (fr) 수학에서 복소 곱셈(영어: complex multiplication)이란 대수적 수체 위에 정의된 특별한 타원 곡선들이 정수의 환보다 더 큰 자기준동형환을 갖는 현상이다. (ko) 虚数乗法(complex multiplication)とは、通常よりも大きな対称性をもつ楕円曲線の理論のことをいう。別のいいかたをすれば、(period lattice)がガウス整数の格子であったり、アイゼンシュタイン整数の格子であったりするような、余剰な対称性を持つ楕円函数の理論である。楕円曲線の高次元化であるアーベル多様体についても同様に大きな対称性をもつ場合があり、これらを扱うのが虚数乗法論である。特殊関数の理論として、そのような楕円函数や多変数複素解析函数のアーベル函数は、大きな対称性をもつことからその関数が多くの等式をみたすことがいえる。特別な点では具体的に計算可能な特殊値を持つ。また虚数乗法は代数的整数論の中心的なテーマであり、円分体の理論をより広く拡張する事を可能にする。虚数乗法は、虚二次体の類体における相互法則、主イデアル定理、分岐の様子を、楕円函数や楕円曲線のことばで具体的に書き表すことを可能とする。ダフィット・ヒルベルトは、楕円曲線の虚数乗法論は数学のみならず、すべての科学の中の最も美しい分野であると言っている。 (ja) In de wiskunde is complexe vermenigvuldiging de theorie van de elliptische krommen waarvan de ring van endomorfismen bestaat uit meer dan de triviale endomorfismen die gegeven worden door vermenigvuldiging met een geheel getal. Meer algemeen is het de theorie in hogere dimensies van abelse variëteiten die in een zekere precieze zin genoeg endomorfismen hebben (dit betekent ruwweg dat de actie van de raakruimte op de identiteitselement van een directe som van eendimensionale modulen is). Anders gezegd is complexe vermenigvuldiging de theorie van de elliptische functies met extra symmetrieën, die voorkomen als het een geheeltallig rooster volgens Gauss of volgens Eisenstein is. (nl) Em matemática, multiplicação complexa é a teoria das curvas elípticas E que tem um maior que os inteiros; e também a teoria de mais altas dimensões das variedades abelianas A tendo suficientes endomorfismos em um certo sentido preciso (isso aproximadamente significa que a ação sobre o espaço tangente de A é uma soma direta de módulos unidimensionais). Visto de outra forma, contém a teoria de funções elípticas com simetrias extras, como são visíveis quando o é o retículo do inteiro de Gauss ou retículo do inteiro de Eisenstein. (pt) In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice. There is also the higher-dimensional complex multiplication theory of abelian varieties A having enough endomorphisms in a certain precise sense, roughly that the action on the tangent space at the identity element of A is a direct sum of one-dimensional modules. (en) En matemáticas, la multiplicación compleja (MC) es la teoría que trata sobre el conjunto de curvas elípticas E sobre el que se define un más grande que el correspondiente a los números enteros; y también la teoría en dimensiones más altas de A que poseen suficientes endomorfismos en un cierto sentido preciso (significa aproximadamente que la acción en el espacio tangente sobre el elemento neutro de A es una suma directa de módulos unidimensionales). Dicho de otra manera, contiene la teoría de las funciones elípticas con simetrías adicionales, como son visibles cuando el par fundamental de períodos coincide con el retículo entero gaussiano o con el retículo entero de Eisenstein. (es) In matematica la moltiplicazione complessa (spesso abbreviato con CM, cioè Complex Multiplication) è la teoria delle curve ellittiche che hanno strettamente più grande di ed è anche la teoria delle varietà abeliane che hanno abbastanza endomorfismi in un senso più specifico (informalmente se l'azione dello spazio tangente sull'elemento identità della varietà abeliana è una somma diretta di moduli di dimensione uno). (it)
foaf:depiction
dct:subject	Abelian varieties Class field theory Elliptic functions
Wikipage page ID	537183 (xsd:integer)
Wikipage revision ID	1114695582 (xsd:integer)
Link from a Wikipage to another Wikipage	Carl Friedrich Gauss Princeton University Press Endomorphism ring Module (mathematics) David Hilbert Hodge conjecture Elliptic curves Shimura's reciprocity law Wiles's proof of Fermat's Last Theorem Mathematics Elliptic function Eisenstein integer Eisenstein series Elliptic curve Frobenius map Gaussian integer Gotthold Eisenstein Order (ring theory) Roots of unity Leopold Kronecker Singular curve Complex multiplication of abelian varieties Ideal class group Identity element Tangent space Automorphism Torus Weierstrass elliptic function Galois extension Galois group Heegner point Lattice (group) Local field Algebraic number Algebraic number theory Cyclotomic field Finite field Abelian function Direct sum of modules Graduate Texts in Mathematics Hilbert's twelfth problem Hilbert class field Unique factorization domain Heegner number J-invariant Abelian extension Abelian variety Abelian varieties Class field theory Elliptic functions Bulletin of the AMS Polynomial Special function Springer Science+Business Media Class field theory Field extension Maximal abelian extension Integer Rational number Complex coordinate space PlanetMath Quaternion algebra Modular forms Kronecker Jugendtraum Drinfeld shtuka Springer-Verlag Elliptic modular function Langlands philosophy

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