Consider a simple abelian variety $A/\mathbb{C}$ with sufficiently many CMs by $\mathcal{O}$, where $\mathcal{O}$ is an order in a CM field $K$. Specifically, $K$ is a CM field of degree $2g$, where $g = \dim A$. Let $(K,\, \Phi)$ be the CM type. It can be shown that there is a field $F$ such that $A$ is defined and has smCM over $F$.

*Is there an explicit field $F_0(\mathcal{O},\, \Phi)$ such that $A$ is defined over $F_0$, with CM over $F_0$?*

As motivation: If $A$ is an elliptic curve, $K$ an imaginary quadratic field, and $\mathcal{O}$ is the ring of integers in $K$, then the answer is yes: $A$ is defined with CM over the Hilbert class field $H(K)$, and moreover there is no better field of definition, so $F_0$ can be taken to be $H(K)$. It is moreover my understanding that if $\mathcal{O}$ is a different order, then $F_0$ can be taken to be the Hilbert class field of the order $\mathcal{O}$, though this is something I heard and might be misinformed.

My question, then, is whether this extends to higher-dimensional abelian varieties. It's known that such an abelian variety can be defined over *some* number field, but the arguments I've seen in this direction are of the form "if $A$ is defined over $\mathbb{C}$, then it can be defined over a finite-type ring over $\mathbb{Q}$, so reduce modulo a maximal ideal and check that it works." There is, on the other hand, an old result of Taniyama-Shimura that says: if $A$ has CM type $(K,\, \Phi)$ and $A$ is defined over $F$, then $F$ contains the reflex field $E(K,\, \Phi)$.

So, can the result:
*An elliptic curve $A/\mathbb{C}$ with CM by $\mathcal{O}$ can be defined over the Hilbert class field of $\mathcal{O}$*
be extended, in some form, to higher-dimensional abelian varieties? And if so, in what form?

Thanks in advance.

Complex multiplication of abelian varieties and applications to number theory). $\endgroup$ – Vesselin Dimitrov Aug 6 '14 at 16:43notfields of definition. Very far from it. This is related to one reason that coarse moduli spaces (rather than moduli stacks) generally suck: their rational points have very limited meaning. If you don't allow isogeny change over $\mathbf{C}$ then it is false that a field of definition can be predicted by the CM type which is isogeny-invariant. It is a very subtle problem beyond dimension 1 to nail down a specific field of definition for a specific CM abelian variety (equipped with its CM-structure) without moving around in the isogeny class. $\endgroup$ – user27920 Aug 7 '14 at 0:22