Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 23F (Riemann Surfaces)

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.

Let $q\in {\mathbb{C}}_{\infty }$ and let $G_q$ be its stabilizer subgroup. Choose a sufficiently small disc $U$ about $q$ such thatPart (a) says that $G_q$ is cyclic and supplies a coordinate in which $U\to G_q\backslash U$ is $z\mapsto z^{|G_q|}$. Since the global orbit map identifies no additional points within $U$, this is also a chart on a neighbourhood of the orbit $Gq$ in

Use these quotient charts at every orbit. On overlaps, a local inverse branch of one power map followed by a group translate and the other power map gives the transition function; it is holomorphic. The charts therefore define a conformal structure on $R$, and the local expressions $z\mapsto z^{|G_q|}$ show directly that the quotient map ${\mathbb{C}}_{\infty }\to R$ is holomorphic. This is the finite conformal quotient of a Riemann surface construction.

Let $\zeta =e^{2\pi i/n}$ and define Möbius transformationsThey satisfyand the $2n$ maps $r^k$ and $s{r}^k$ are distinct. They therefore give a faithful action of the dihedral group $D_{2n}$ on the Riemann sphere.

The rational functionis invariant under both $r$ and $s$, so it is constant on every orbit. Conversely, for nonzero finite $z_1,z_2$, put $u=z_1^n$ and $v=z_2^n$. Equality of the function values givesIf $u=v$, then $z_2={\zeta }^k{z}_1=r^k{z}_1$ for some $k$. If $uv=1$, then $z_2={\zeta }^k/z_1=r^{k}s{z}_1$ for some $k$. Finally, zero and infinity both map to infinity and are interchanged by $s$. HenceThis is the orbit-separating invariant for the standard dihedral action on the Riemann sphere.