Fix a CM-field $K$ of degree $2g$, and a natural number $n$ which is a multiple of $g$. Write

$\tau_1, \tau_2, \ldots, \tau_g, \rho \tau_1, \rho \tau_2, \ldots, \rho \tau_g$

for the different embeddings of $K$ into $\mathbb{C}$, where $\rho$ denotes complex complex conjugation. Let $\Phi$ be an $n$-dimensional complex representation of $K$ in the form

$\Phi = \bigoplus_{\nu = 1}^g (r_\nu \cdot \tau_\nu \oplus s_\nu \cdot \rho \tau_\nu)$ where $r_1 + s_1 = r_2 + s_2 = \cdots = r_g + s_g = \frac{n}{g}$.

Consider the moduli space of pairs $(A, \theta)$ where $A$ is an $n$-dimensional abelian variety and $\theta : K \hookrightarrow End(A) \otimes_{\mathbb{Z}} \mathbb{Q}$ is an injection such that the $n$-dimensional complex representation of $K$ defined by the tangent space to the identity of $A$ is isomorphic to $\Phi$.

In terms of $(r_1, s_1, \ldots, r_g, s_g)$, what is the minimal field over which this moduli space is defined? Moreover, what is the minimal extension of this field over which all of its irreducible components are defined?

EDIT: As pointed out by Kevin Buzzard below, what I really want is the minimal field not just where the moduli space is defined, but where the corresponding moduli functor is also defined. Moreover (as pointed out by Keerthi below), we should probably also fix a polarization of $A$, in order to guarantee that this moduli space exists.

spaceis defined, rather than the minimal field over which thefunctoris defined? Say I have a natural moduli problem which onlymakes sensefor schemes over $F$, $F$ some number field (e.g. it mentions something like "scheme plus action of $F$ plus blah such that the two induced $F$-actions on (something) agree" (one action coming from the fact that the scheme is defined over $F$). Say this problem happens to be represented by projective 1-space over $F$. Then the functor is defined over $F$ but... – Kevin Buzzard Feb 12 '11 at 9:38spaceis defined over the rationals, and indeed there will probably be several non-isomorphic spaces over the rationals whose pullback to $F$ give projective 1-space, and there is no way of determining which, if any, is the "right" model for the space---until you have extended the functor. – Kevin Buzzard Feb 12 '11 at 9:40