Let $E/ \Bbb{C}$ be an elliptic curve which has complex multiplication over a number field $K$.


Then it is widely known that $j(E) \in \overline { \Bbb{Z}}$.


What is the known generalization of this statement to abelian varieties of arbitrary dimension?



Generalization of $j(E) \in \overline { \Bbb{Z}}$ to abelian varieties of arbitrary dimension
The $j$ invariant gives an isomorphism over $\mathbb Z$,
$$j:\mathcal A\_1\to\mathbb A^1,$$
of the moduli space of elliptic curves. So $j(E)\in\overline{\mathbb Z}$ can be interpreted as saying that $\langle E\rangle\in \mathcal A\_1(\overline{\mathbb Z})$, where I've written $\langle E\rangle$ for the isomorphism class of $E$. As noted by Dror, there are Igusa invariants which (largely) do the same thing for $\mathcal A\_2$. But in general, probably the right interpretation is that one should look at the moduli space $\mathcal A\_g/\mathbb Z$, and then it is a theorem that an abelian variety $A$ that has complex multiplication satisfies
$$ \langle A\rangle \in \mathcal A\_g(\overline{\mathbb Z}). $$
Another useful interpretation is that the elements of $\mathcal A\_g(\overline{\mathbb Z})$ have potential good reduction, so in particular this is true for CM abelian varieties. There is also the criterion of Neron-Ogg-Shafarevich that you might find relevant; see
*Serre, Jean-Pierre; Tate, J.*, [**Good reduction of abelian varieties**](http://dx.doi.org/10.2307/1970722), Ann. Math. (2) 88, 492-517 (1968). [ZBL0172.46101](https://zbmath.org/?q=an:0172.46101).


