Editing Complex multiplication (section)

{{Short description|Theory of a class of elliptic curves}}
{{about|a topic in the theory of [[elliptic curves]]|information about multiplication of complex numbers|complex numbers}}
In [[mathematics]], '''complex multiplication''' ('''CM''') is the theory of [[elliptic curve]]s ''E'' that have an [[endomorphism ring]] larger than the [[integer]]s.{{sfn|Silverman|2009|p=69|loc=Remark 4.3}} Put another way, it contains the theory of [[elliptic function]]s with extra symmetries, such as are visible when the [[period lattice]] is the [[Gaussian integer]] [[Lattice (group)|lattice]] or [[Eisenstein integer]] lattice.

It has an aspect belonging to the theory of [[special function]]s, because such elliptic functions, or [[abelian function]]s of [[several complex variables]], are then 'very special' functions satisfying extra identities and taking explicitly calculable special values at particular points. It has also turned out to be a central theme in [[algebraic number theory]], allowing some features of the theory of [[cyclotomic field]]s to be carried over to wider areas of application. [[David Hilbert]] is said to have remarked that the theory of complex multiplication of elliptic curves was not only the most beautiful part of mathematics but of all science.<ref>{{Citation
| last=Reid
| first=Constance
| author-link=Constance Reid
| title=Hilbert
| publisher=Springer
| isbn=978-0-387-94674-0
| year=1996
| page=[https://archive.org/details/hilbert0000reid/page/200 200]
| url=https://archive.org/details/hilbert0000reid/page/200
}}</ref>

There is also the [[Complex multiplication of abelian varieties|higher-dimensional complex multiplication theory]] of [[abelian variety|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 modules|direct sum]] of one-dimensional [[module (mathematics)|modules]].