complex multiplication

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

The following theorem implies that every endomorphism is also agroup endomorphism:

Theorem 1

Let $E_{\mathrm{1}}\mathrm{,}{E}_{\mathrm{2}}$ be elliptic curves, and let $\varphi \mathrm{:}{E}_{\mathrm{1}}\mathrm{\to }{E}_{\mathrm{2}}$ be a regular map such that$\varphi \mathit{}\mathrm{(}{O}_{E_{\mathrm{1}}}\mathrm{)}\mathrm{=}{O}_{E_{\mathrm{2}}}$. Then $\varphi$ is also a group homomorphism,i.e.

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

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

If $\mathrm{End}(E)$ is isomorphic (as a ring) to an order (http://planetmath.org/OrderInAnAlgebra) $R$ in a quadratic imaginaryfield $K$ then we say that the elliptic curve E has complexmultiplication by $K$ (or complex multiplication by $R$).

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

$$[n]:E\to E,[n]P=n\cdot P$$

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

Example: Fix $d\in \mathbb{Z}$. Let $E$ be the ellipticcurve 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 multiplicationby $n$ maps, $\mathrm{End}(E)$ contains a genuine new element:

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

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