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Let $A$ be a complex abelian variety such that $\mathrm{End}(A)$ is strictly bigger than $\mathbb{Z}$.

Question: Is is true that $A$ is defined over $\overline{\mathbb{Q}}$?

This is of course what happens for elliptic curves.

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Dear Vima, Not in general, no. E.g. if $E$ is a fixed CM elliptic curve, and $E'$ is an arbitrary elliptic curve (e.g. one one not defined over $\overline{\mathbb Q}$), then $E\times E'$ is an abelian variety not defined over $\overline{\mathbb Q}$, with extra endomorphisms. There are also many less trivial examples than this of positive dimensional families of ab. vars. with extra endomorphisms. (These are basically what PEL-type Shimura varieties are.) Regards, –  Emerton Jun 28 '13 at 12:35
    
So under which conditions is that true? Is that the case if $End(A) \otimes_\mathbb{Z} \mathbb{Q}$ is a CM field of degree twice the dimension? –  vima Jun 28 '13 at 12:43
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