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

The modular curve isGive 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 $\sigma \in S{L}_2(\mathbb{Z})$ and having width $h$, a punctured neighborhood is described byand adjoining $q_h=0$ fills in the cusp. These charts make $X(\Gamma )$ a compact Riemann surface.