complex multiplication

Let $E$ be an elliptic curve. The endomorphism ring of $E$ , denoted $\operatorname{End}(E)$ , is the set of all regular maps $\phi\colon E\to E$ such that $\phi(O)=O$ , where $O\in E$ is the identity element for the group structure of $E$ . Note that this is indeed a ring under addition ( $(\phi+\psi)(P)=\phi(P)+\psi(P)$ ) and composition of maps.

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

Theorem 1

Let $E_{1},E_{2}$ be elliptic curves, and let $\phi\colon E_{1}\to E_{2}$ be a regular map such that $\phi(O_{E_{1}})=O_{E_{2}}$ . Then $\phi$ is also a group homomorphism, i.e.

$\forall P,Q\in E_{1},\ \phi(P+_{E_{1}}Q)=\phi(P)+_{E_{2}}\phi(Q).$

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

If $\operatorname{End}(E)$ is isomorphic (as a ring) to an order (http://planetmath.org/OrderInAnAlgebra) $R$ 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: $\operatorname{End}(E)$ always contains a subring isomorphic to $\mathbb{Z}$ , formed by the multiplication by n maps:

$[n]\colon E\to E,\quad[n]P=n\cdot P$

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

Example: Fix $d\in\mathbb{Z}$ . Let $E$ be the elliptic curve defined by

$y^{2}=x^{3}-dx$

then this curve has complex multiplication by $\mathbb{Q}(i)$ (more concretely by $\mathbb{Z}(i)$ ). Besides the multiplication by $n$ maps, $\operatorname{End}(E)$ contains a genuine new element:

$[i]\colon E\to E,\quad[i](x,y)=(-x,iy)$

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

References

1 James Milne, Elliptic Curves, online course notes. http://www.jmilne.org/math/CourseNotes/math679.htmlhttp://www.jmilne.org/math/CourseNotes/math679.html
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