Endomorphism lattice (reviewed)

Let $A$ be an abelian variety over a field $F$. Given a field extension $F' \supseteq F$, there is a natural inclusion of endomorphism rings $\End(A)\subseteq\mathrm{End}(A_{F'})$, where $A_{F'}$ denotes the base change of $A$ to $F'$.

Let $K$ be the endomorphism field of $A$. The set of endomorphism rings $\mathrm{End}(A_{F'})$ over subextensions $K \supseteq F' \supseteq F$ forms a lattice under inclusion, called the endomorphism lattice of $A$ over $F$.