# Endomorphisms of abelian varieties with real multiplication

Let us work over $\mathbb{C}$ to make life easier.

I've came across to the following definition. Let $F$ be a totally real number field of degree $g$, with ring of integers $\mathcal{O}_F$. An abelian variety $A$ of dimension $g$ has real multiplication by $\mathcal{O}_F$ if there is a ring embedding $\mathcal{O}_F \to {\rm End}(A)$.

Question: Is it always the case that $F$ maps into the center of ${\rm End}^0(A)$? What if $A$ is simple?

In general, can the structure of ${\rm End}^0(A)$ be completely understood?

• Under that definition, the answer is no, as the example of an abelian surface with quaternion algebra multiplication already shows. – Will Sawin Apr 18 '18 at 20:09
• The structure of $\mathrm{End}^0(A)$, for $A$ simple, is described by the Albert classification of division algebras with involution. See Mumford's book on abelian varieties, or §12 of math.ru.nl/~bmoonen/research.html#bookabvar – jmc Apr 18 '18 at 20:33