# field of definition of CM abelian varieties

When $$A$$ is a CM abelian variety of dimension $$1$$ (i.e., an elliptic curve), then we have a result that if it has CM by a maximal order then it has a model over a number field $$F$$ where $$F$$ is the Hilbert class field of the CM field if $$A$$ has CM by the maximal order and is otherwise a ray class field. In particular, if $$A$$ has CM by a non-maximal order then it has a "strictly larger" field of definition. (Experts, please excuse some sloppiness here.)

Is the analogous result true in general? I.e., is it true that if we take two CM abelian varieties $$A$$ and $$A^\prime$$ with CM by $$(K,\Phi)$$ but such that $$A$$ has CM by the maximal order $$\mathcal{O}_K\subseteq K$$ and $$A^\prime$$ has CM by a non-maximal order $$\mathcal{O}\subseteq K$$ that for the analogous model of $$A^\prime$$ over $$F^\prime$$ and $$A$$ over $$F$$ we must have that $$[F^\prime:F]>1$$?

• For an elliptic curve $E/F$ with CM endomorphisms defined over $F$ : there is some $n$ such that it has CM by $\mathbb{Z}+n\mathcal{O}_K$, let $H = \{ P \in E, \forall a \in n \mathcal{O}_K, [a] P = 0\}$ then $E/H$ has CM by $\mathcal{O}_K$. The isogeny $E \to E/H$ is defined over $F$, so $F$ contains the field where curves with CM by $\mathcal{O}_K$ are defined. – reuns Jan 29 at 0:51