Endomorphism ring of an elliptic curve (reviewed)

The endomorphism ring $\End(E)$ of an elliptic curve \(E\) over a field $K$ is the ring of all endomorphisms of \(E\) defined over $K$. For endomorphisms defined over extensions, we speak of the geometric endomorphism ring of $E$.

For elliptic curves defined over fields of characteristic zero, this ring is isomorphic to \(\Z\), unless the curve has complex multiplication (CM) defined over the ground field, in which case the endomorphism ring is an order in an imaginary quadratic field; for curves defined over \(\Q\), this order is one of the 13 orders of class number one.

$\End(E)$ always contains a subring isomorphic to $\Z$, since for $m\in\Z$ there is the multiplication-by-$m$ map $[m] \colon E\to E$.

This is a special case of the endomorphism ring of an abelian variety.