Shimura Varieties actual varieties for any level? Since I have not been answered on my other thread Shimura varieties/Double coset for non-neat level so far and I can't work with this uncertainty, I'll just put down the most important questions here. I would still be grateful if someone would take time for the other thread. Let $(G,X)$ be a Shimura datum. Are the level $K$ varieties $\text{Sh}_K(G,X)$ also (quasi-projective) varieties if $K$ is not sufficiently small (in which case they are normal)? Do the $X_K=\text{Sh}_K(G,X)$ also have models over a number field if $K$ is not sufficiently small? Does the tower of Shimura varieties $\{X_K\}$ have a model over $E$ even if $K$ runs through all open compact $K$? Now let $D$ be a general Hermitian symmetric domain and $\Gamma \subset \text{Is}^+(D,g)$ a discrete subgroup of the holomorphic isometry group of $D$. Bailey-Borel says $\Gamma\backslash D$ is a normal quasi-projective variety over $\mathbb{C}$ for any $\Gamma$ (not necessarily torsionfree). Does it have a model over a number field?
 A: First as you said Bailey-Borel theorem says that $D/\Gamma$ is analyticification of a quasi-projective smooth variety for any torison free $\Gamma$ and a quasi-projective normal variety for any $\Gamma$.
you can understand all $Sh(G,X)_K$ as varieties, and they all have canonical models over reflex field.(although sometimes it is better to look at the $Sh(G,X)_K$ for big $K$ as a stack because the variety is only a coarse solution to moduli problem you want to consider).
all the connected components have a model over some number field so $D/\Gamma$ always have a model over some number field for any $\Gamma$ that comes from a congurance subgroup of some $G$ such that $G^{ad}=Hol(D)$.maybe for any $\Gamma$ you can find a model over a number field but this model is not in any way canonical and after all the interesting point about canonical models is that you have a model over a fixed number field otherwise you only want a model over $\bar{\mathbb{Q}}$.
*just to clarify what I mean by look at big level shimura variety as a stack: you can write $Sh(G,X)_K=Sh(G,X)_{K'}/\frac{K}{K'}$ for a neat group $K'$, this quotient is represented by a normal quasi-projective variety in the category of schemes(because $\frac{K}{K'}$ is a finite group) but sometimes it is better to look at the quotient stack which could be equal to that normal variety or not.
