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:

If $\operatorname{End}(E)$ 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: $\operatorname{End}(E)$ 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).

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.