complex multiplication

Let E be an elliptic curveMathworldPlanetmath. The endomorphism ringMathworldPlanetmathPlanetmath of E, denoted End(E), is the set of all regular maps ϕ:EE such that ϕ(O)=O, where OE is the identity elementMathworldPlanetmath for the group structure of E. Note that this is indeed a ring under addition ((ϕ+ψ)(P)=ϕ(P)+ψ(P)) and composition of maps.

The following theorem implies that every endomorphismPlanetmathPlanetmath is also a group endomorphismPlanetmathPlanetmath:

Theorem 1

Let E1,E2 be elliptic curves, and let ϕ:E1E2 be a regular map such that ϕ(OE1)=OE2. Then ϕ is also a group homomorphism, i.e.


[Proof: See [2], Theorem 4.8, page 75]

If End(E) is isomorphic (as a ring) to an order ( R in a quadratic imaginary field K then we say that the elliptic curve E has complex multiplication by K (or complex multiplication by R).

Note: End(E) always contains a subring isomorphic to , formed by the multiplication by n maps:


and, in general, these are all the maps in the endomorphism ring of E.

Example: Fix d. Let E be the elliptic curve defined by


then this curve has complex multiplication by (i) (more concretely by (i)). Besides the multiplication by n maps, End(E) contains a genuine new element:


(the name complex multiplication comes from the fact that we are “multiplying” the points in the curve by a complex numberMathworldPlanetmathPlanetmath, i in this case).


  • 1 James Milne, Elliptic Curves, online course notes.
  • 2 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
  • 3 Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.
  • 4 Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey, 1971.
Title complex multiplication
Canonical name ComplexMultiplication
Date of creation 2013-03-22 13:41:35
Last modified on 2013-03-22 13:41:35
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 15
Author alozano (2414)
Entry type Definition
Classification msc 11G05
Related topic EllipticCurve
Related topic KroneckerWeberTheorem
Related topic OrderInAnAlgebra
Related topic ArithmeticOfEllipticCurves
Defines complex multiplication
Defines endomorphism ring