# Generalization of $j(E) \in \overline { \Bbb{Z}}$ to abelian varieties of arbitrary dimension

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?

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

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.
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.)