Let $(G, X)$ be a Shimura datum (so $G$ is a reductive group over $\mathbb Q$). For neat compact open $K \leq G(\mathbb A_f)$, we consider associated Shimura variety $M=Sh_K(G,X)$ with
$M(\mathbb C)= G(\mathbb Q) \backslash X \times G(\mathbb A_f) / K.$
I am interested in the case $\dim M=1$, so $M$ is a Shimura curve. Note in general, $M$ is not of Hodge type and there is no moduli description.
For example, we have Shimura curves from quaternion algebras (split at exactly one real place) over a totally real number field.
From the general theory, we know that $M$ is defined over $\mathbb Q^{alg}$. Is there a simpler proof of this fact for curves?
From Belyi's theorem on curves, we know a (connected) complex smooth projective curve $S$ is defined over $\mathbb Q^{alg}$ iff there is a (so called Beyli map) holomorphic map from $S(\mathbb C) \to \mathbb P^1(\mathbb C)$ ramified only over three points.
In general, $M$ is far from connected (over $\mathbb C$). The question is, do we know a direct (explicit) construction of Beyli maps for (a connected component) of $M(\mathbb C)$? What will be the minimal degree of such maps (in terms of $K$)? Could we find Beyli maps over reflex fields?
I only know some examples of modular curves (with small levels). As the transition map between different $K_1 \leq K_2$ is finite etale (over $\mathbb C$), for our questions it would suffice to work with $M$ where $K$ is very large. But to make sure $M$ is not stacky, non-maximal $K$ seems necessary.
