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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?

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    $\begingroup$ I think this is probably in Fite-Kedlaya-Rotger-Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arxiv.org/abs/1110.6638. They classify Sato-Tate groups for abelian surfaces, and explain the relation to Mumford-Tate $\endgroup$
    – Tim Dokchitser
    Commented Dec 8, 2017 at 19:44

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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 .

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  • $\begingroup$ Thank you for this amazing reference. So, for instance, the Mumford-Tate group of an abelian surface with real multiplication is always of dimension 7? And for an abelian surface with quaternionic multiplication (split at $\infty$) it's always of dimension 4? $\endgroup$
    – user94490
    Commented Dec 10, 2017 at 17:56
  • $\begingroup$ You are welcome. Yes, that's correct. $\endgroup$
    – Yuri Zarhin
    Commented Dec 10, 2017 at 18:37

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