= Solution
The <modular curve> is
$$
X(\Gamma)=\Gamma\backslash
\bigl(\mathfrak h\cup\mathbb P^1(\mathbb Q)\bigr).
$$
Give it the quotient topology on $\Gamma\backslash\mathfrak h$ and adjoin one end for each <cusp of a modular group>. At a point with trivial effective stabilizer, the quotient map identifies a sufficiently small disk with a chart. At an elliptic point with cyclic stabilizer of order $e$, choose a disk coordinate $z$ centered there; then $z^e$ descends to a quotient coordinate. At a cusp represented by $\sigma\in SL_2(\mathbb Z)$ and having width $h$, a punctured neighborhood is described by
$$
q_h=e^{2\pi i\sigma^{-1}\tau/h},
$$
and adjoining $q_h=0$ fills in the cusp. These charts make $X(\Gamma)$ a compact <Riemann surface>.
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