For elliptic curves, one may easily compute Mumford-Tate groups; there are just two cases:


1) $E$ has no complex multiplication, and the Mumford-Tate group of $E$ is $GL\_2$


2) $E$ has complex multiplication by a field $k$ and the Mumford-Tate group $E$ is a torus (of dimension two) in $GL\_2$ induced by $k$.


In any case, the list of possible dimensions of Mumford-Tate groups of elliptic curves is $\{2,4\}$.


Is it possible to explicitly classify all the Mumford-Tate groups of abelian surfaces? What is the list of possible dimensions of Mumford-Tate groups in this case?



Mumford-Tate groups of abelian surfaces
The list of of all possible Hodge (special Mumford-Tate) groups of complex abelian varieties up to dimension 4 (and for simple abelian varieties up to dimension 5) is contained in <https://arxiv.org/pdf/math/9901113.pdf> <https://link.springer.com/article/10.1007%2Fs002080050333> .


