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Let $D$ be a division algebra over a number field $K$, and consider abelian varieties $A$ over the complex numbers, of dimension $g$ with an action of (an order of) $D$. Is it known when this set is non-empty, and if so what is the dimension of this space?

I always thought this was a solved question but glancing over the literature it seems like it might be tricky. Is there a good reference?

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    $\begingroup$ Did you look at Chapter 9 of Birkenake-Lange's book Complex Abelian Varieties? Its title is Moduli spaces of abelian varieties with endomorphism structure. $\endgroup$
    – Francesco Polizzi
    Mar 6, 2017 at 9:36
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    $\begingroup$ Nope. I found some papers of Oort (say " A construction of an abelian variety with a given endomorphism algebra" eudml.org/doc/89907) which seemed to suggest the problem was tricky. Does that chapter answer the question completely? $\endgroup$
    – jacob
    Mar 6, 2017 at 9:50
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    $\begingroup$ Maybe not completely, but there are several interesting examples. $\endgroup$
    – Francesco Polizzi
    Mar 6, 2017 at 9:59

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