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>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 137 / 3 / a / Solution

A cusp of a finite-index subgroup $\Gamma\leq SL_2(\mathbb Z)$ is an orbit in $\Gamma\backslash\mathbb P^1(\mathbb Q)$. If $\sigma\in SL_2(\mathbb Z)$ sends infinity to a cusp, behavior there is studied through $f|_k\sigma$.


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 Incoming links: Cusp of a modular group