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I've got a few questions related to a paper by Lei Yang - "Exotic Arithmetic Structure on the First Hurwitz Triplet"

We know that there are exactly three Hurwitz curves of genus 14. Lets call them $X_1,X_2$, and $X_3$. Let $G_{2,3,7}$ be the triangle group of orientation-preserving transformations generated by reflections of a given hyperbolic triangle with angles $\pi/2,\pi/3,\pi/7$. Ie, $$G_{2,3,7} = \langle \sigma_2,\sigma_3,\sigma_7 : \sigma_2^2 = \sigma_3^3 = \sigma_7^7 = \sigma_2\sigma_3\sigma_7 = 1\rangle$$

Then we know that any Hurwitz curve can be realized as $\mathcal{H}/\Gamma$, where $\mathcal{H}$ is the upper half plane, and $\Gamma$ is its fundamental group, which is necessarily a normal subgroup of $G_{2,3,7}$. In this case, the automorphism group of the Hurwitz curve is $G_{2,3,7}/\Gamma$.

The three Hurwitz curves can be viewed as Shimura curves as described in the paper, but also in

In particular, each of $X_1,X_2,X_3$ has $PSL(2,13)$ as its automorphism group.

Later on Lei Yang goes on to give a 6 dimensional projective representation of $PSL(2,\mathbb{Z})$ (ie, a map $PSL(2,\mathbb{Z})\rightarrow PGL(6,\mathbb{Q}(\zeta_{13}))$), where the image is isomorphic to $PSL(2,13)$. In this particular case, the kernel $G$ ends up being a noncongruence subgroup of $PSL(2,\mathbb{Z})$, resulting in a compactified modular curve $\overline{\mathcal{H}/G}$ of genus 14 which has automorphism group $PSL(2,13)$, and hence this noncongruence modular curve must be one of the Hurwitz curves $X_1,X_2$, or $X_3$.

$$\textbf{MY QUESTION:}$$ In what sense does the Shimura curve construction mean that the action of $PSL(2,13)$ on $X_1,X_2,X_3$ is "defined over $\mathbb{Q}(\zeta_7)$" ?

Similarly, in the noncongruence case, in what sense is the action of $PSL(2,13)$ on the noncongruence modular curve $\overline{\mathcal{H}/G}$ defined over $\mathbb{Q}(\zeta_{13})$?

I suppose my real question is, given a description of a Fuchsian group $\Gamma$ (for example the one from the wiki article), how would you come up with an algebraic model of $\mathcal{H}/\Gamma$? In particular, is there a nice way to come up with an algebraic model that makes it obvious why the actions of $PSL(2,13)$ on the two versions of these Hurwitz curves (shimura and noncongruence) are defined over $\mathbb{Q}(\zeta_7)$ and $\mathbb{Q}(\zeta_{13})$ ?


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