Suppose $\mathcal H$ is the upper half plane, and $\Gamma\subset PSL_2(\mathbb Z)$ is a finite index subgroup. Then by Belyi's theorem we know that $\Gamma\setminus \mathcal H$ is an algebraic curve defined over a number field $E$. I want to ask is there any research on etale $\mathbb Q_p$-local system of this curve?

For example there's a natural class of local systems arises from $\mathbb Q$-representations of $PSL_2(\mathbb Q)$ (just as Shimura varieties), and can we talk anything about the galois representation of the local system at a point?

For Shimura varieteis, using special point and artin map we see the representation factors through $Gal(E^{ab}/E)$. But for non-congruent $\Gamma$, adelic tricks fails. Do we have some other ways to handle it?

Any related references would be appreciated.

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.