Endomorphism algebra (reviewed)

The endomorphism algebra of an abelian surface $A$ over a field $K$ is the $\Q$-algebra $\End(A) \otimes \Q$. This is a special case of the endomorphism algebra of an abelian variety.

When $A$ is an abelian surface over $\Q$, there are five possibilities for $\End(A)\otimes\Q$:

1. $\Q$;
2. a real quadratic field (in which case $A$ has real multiplication, denoted RM);
3. an imaginary quadratic field (in which case $A$ has complex multiplication, denoted CM);
4. $\Q\times \Q$;
5. $\mathrm{M}_2(\Q)$;

The first three cases occur when $A$ is simple, while the last two cases occur when $A$ is isogenous to a product of elliptic curves $E_1\times E_2$ over $\overline K$. Which of the last two cases occurs depends on whether $E_1$ and/or $E_2$ are isogenous or not.