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Complex Multiplication

Let be an elliptic curve. The endomorphism ring of , 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 . 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 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.

$\displaystyle \forall P,Q \in E_1,\ \phi(P +_{E_1} Q)=\phi(P)+_{E_2}\phi(Q).$

[Proof: See $\cite{silverman}$ , Theorem 4.8, page 75]

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

Note: $\operatorname{End}(E)$ always contains a subring isomorphic to $\mathbb{Z}$ , formed by the multiplication by n maps:

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

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

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

$\displaystyle y^2=x^3-dx$

then this curve has complex multiplication by $\mathbb{Q}(i)$ (more concretely by $\mathbb{Z}(i)$ ). Besides the multiplication by

maps, $\operatorname{End}(E)$ contains a genuine new element:

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

(the name complex multiplication comes from the fact that we are “multiplying” the points in the curve by a complex number,

in this case).

Bibliography

1: James Milne, Elliptic Curves, online course notes. http://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.

1	Types Of Homomorphisms
1	Isomorphism
1	Curve
1	Point
1	Complex Number
1	Proof
1	Structures And Satisfaction
1	Function
1	Set
1	Fix
1	Implication
1	Ring
1	Subring
1	Multiplication
1	Addition
1	Identity Element
1	Group Homomorphism
1	Group
4	Elliptic Curve
5	Imaginary Quadratic Field
8	Order In An Algebra
12	Regular Map