Suppose $\mathcal H$ is the upper half plane, and $\Gamma\subset \operatorname{PSL}_2(\mathbb Z)$ is a finite index subgroup. Then by Belyi's theorem we know that $\Gamma\backslash\mathcal H$ is an algebraic curve defined over a number field $E$. I want to ask is there any research on étale $\mathbb Q_p$-local system of this curve?
For example there's a natural class of local systems arises from $\mathbb Q$-representations of $\operatorname{PSL}_2(\mathbb Q)$ (just as Shimura varieties), and can we talk anything about the Galois representation of the local system at a point?
For Shimura varieties, using special point and Artin map we see the representation factors through $\operatorname{Gal}(E^\text{ab}/E)$. But for non-congruent $\Gamma$, adelic tricks fails. Do we have some other ways to handle it?
Any related references would be appreciated.
\setminusspaces differently from a left quotient; compare $\Gamma\setminus \mathcal H$\Gamma\setminus \mathcal Hto $\Gamma\backslash\mathcal H$\Gamma\backslash\mathcal H. I'm pretty sure you wanted the latter, so I edited accordingly. $\endgroup$