Geometric endomorphism algebra (reviewed)

The geometric endomorphism algebra of an abelian surface $A$ over a field $K$ is the $\Q$-algebra $\End(A_{\overline{K}}) \otimes \Q$, the endomorphism algebra of the base change $A_{\overline{K}}$ of $A$ to an algebraic closure $\overline{K}$ of $K$. This is a special case of the geometric endomorphism algebra of an abelian variety.

When $A$ is an abelian surface over a number field $K$, there are nine possibilities for $\End(A_{\overline K})\otimes\Q$:

1. $\Q$;
2. a real quadratic field (in which case $A$ has real multiplication, denoted RM);
3. a quartic CM field (in which case $A$ has complex multiplication, denoted CM);
4. a non-split quaternion algebra over $\Q$ (in which case $A$ has quaternionic multiplication, denoted QM);
5. $\Q\times \Q$;
6. $F\times \Q$, where $F$ is a quadratic CM field (denoted $\mathrm{CM} \times \Q$);
7. $F_1\times F_2$, where $F_1$ and $F_2$ are distinct quadratic CM fields (denoted $\mathrm{CM} \times \mathrm{CM}$);
8. $\mathrm{M}_2(\Q)$;
9. $\mathrm{M}_2(F)$, where $F$ is a quadratic CM field (denoted $\mathrm{M}_2(\mathrm{CM})$).

The first four cases occur when $A$ is geometrically simple, while the last five cases occur when $A_{\overline K}$ is isogenous to a product of elliptic curves $E_1\times E_2$ over $\overline K$. Which of the last five cases occurs depends on whether $E_1$ and/or $E_2$ have complex multiplication, and whether $E_1$ and $E_2$ are isogenous or not.