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:

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

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:

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

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:

(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).