Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 23F / a / Solution

Choose a local coordinate $w$ with $w(p)=0$. Differentiation at $p$ gives a homomorphismIt is injective. Indeed, a nonidentity Möbius transformation fixing $p$ and having derivative one there is parabolic, hence conjugate to a nonzero translation and of infinite order; no such element can belong to the finite group $H$. Every finite subgroup of ${\mathbb{C}}^{\times }$ is a cyclic group of roots of unity, so $H$ is cyclic. Write $n=|H|$.

The averaged coordinatehas derivative one at $p$, so it is a coordinate on a smaller neighbourhood. For $g\in H$, reindexing by $k=hg$ givesAfter shrinking to an $H$-invariant disc $U$, the group therefore acts as all rotations $z\mapsto {\zeta }^{j}z$, where $\zeta$ is a primitive $n$th root of unity. The invariant coordinateidentifies $H\backslash U$ with a disc and gives it a Riemann-surface chart. In these coordinates the quotient map is exactlyThis is the local cyclic quotient of a Riemann surface.