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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?

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    $\begingroup$ For hyperelliptic curves (and their jacobians) there's an analogous statement about the Igusa invariants $\endgroup$ Dec 4, 2022 at 11:58

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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, Ann. Math. (2) 88, 492-517 (1968). ZBL0172.46101.

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The integrality of $j(E)$ means that an elliptic curve $E$ over a number field has potential good reduction everywhere (Deuring).

So, the following assertion may be viewed as a generalization of the integrality of the $j$-invariant of elliptic curves with complex multiplication.

An abelian variety of CM type over a number field has potential good reduction everywhere. (See ``Good reduction of abelian varieties" by Serre and Tate, Ann. of Math. 88 (1968), 492--517.)

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