When $A$ is an abelian surface over $\Q$, there are five possibilities for $\End(A)\otimes\Q$:
- a real quadratic field (in which case $A$ has real multiplication, denoted RM);
- an imaginary quadratic field (in which case $A$ has complex multiplication, denoted CM);
- $\Q\times \Q$;
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.
- Review status: reviewed
- Last edited by Edgar Costa on 2020-10-21 14:35:14