Endomorphism ring (reviewed)

An endomorphism of an abelian variety $A$ over a field $F$ is a homomorphism $\varphi \colon A \to A$ defined over $F$. The set of endomorphisms of an abelian variety $A$ can be given the structure of a ring in which addition is defined pointwise (using the group operation of $A$) and multiplication is composition; this ring is called the endomorphism ring of $A$, denoted $\textrm{End}(A)$.

For endomorphisms defined over an extension of $F$, we instead speak about the geometric endomorphism ring.