---
tag: NRJQP7FXFM
kind: statement
role: theorem
name: Riemann's existence theorem
msc: [30F10, 14H30]
uses: [5K0N8QMV6Q, YARENZRXTD, 0515A6D2Y1, 02GK6HZYBT]
about: [5K0N8QMV6Q, YARENZRXTD]
sources:
  - url: https://en.wikipedia.org/wiki/Riemann%27s_existence_theorem
same_as:
  wikidata: Q134889667
  wikipedia: Riemann's existence theorem
---

## Statement

Let $X$ be a compact connected [[Riemann surface|5K0N8QMV6Q]] and let $B \subset X$ be a finite set.

1. Let $p^\circ \colon Y^\circ \to X \setminus B$ be a finite topological covering map with $Y^\circ$ connected.
Then there is a compact connected Riemann surface $Y$ that contains $Y^\circ$ as the complement of a finite set.
There is a holomorphic [[branched covering|YARENZRXTD]] $p \colon Y \to X$ that extends $p^\circ$.
The pair $(Y, p)$ is unique up to a unique isomorphism over $X$ that is the identity on $Y^\circ$.
2. As a consequence, take $X = \hat{\C}$ and $B = \set{b_1, \ldots, b_n}$, and let $d \ge 1$.
Consider the connected holomorphic branched coverings $Y \to \hat{\C}$ of degree $d$ that are unbranched outside $B$, up to isomorphism over $\hat{\C}$.
Consider the tuples $(\sigma_1, \ldots, \sigma_n)$ in the [[symmetric group|0515A6D2Y1]] $S_d$ with $\sigma_1 \cdots \sigma_n = 1$ that generate a [[transitive|02GK6HZYBT]] subgroup.
Call two such tuples equivalent when they are conjugate by one element of $S_d$.
The coverings correspond one to one to the equivalence classes of such tuples.

## Remarks

For part 1, the complex structure of $X$ pulls back to $Y^\circ$ through the local homeomorphism $p^\circ$.
Near a point $b \in B$, each component of the preimage of a small punctured disc is a punctured disc that covers it like $z \mapsto z^e$, and one fills in the puncture.
Part 2 follows by applying part 1 to the coverings of $\hat{\C} \setminus B$, which correspond to transitive actions of the [[fundamental group|S1XK3HXHFX]].
That group is generated by loops $\gamma_1, \ldots, \gamma_n$ around the points $b_i$ with the single relation $\gamma_1 \cdots \gamma_n = 1$, and $\sigma_i$ is the permutation of the $d$ sheets given by $\gamma_i$.

Example: for $n = 2$, $B = \set{0, \infty}$ and $\sigma_1 = (1\,2\,\cdots\,d)$, $\sigma_2 = \sigma_1^{-1}$, the covering is $z \mapsto z^d$.

Since every compact Riemann surface carries enough meromorphic functions to be an algebraic curve, every such datum is realized by a polynomial equation $F(z, w) = 0$.

## Explanations

### L1 human

The theorem says that one can build Riemann surfaces from purely combinatorial data.
To describe a surface lying over the sphere with $d$ sheets, one only needs to say how the sheets are permuted when one walks around each branch point.
Any choice of permutations that multiply to the identity and connect all sheets gives a surface, and only one.

### L2 human

This is the link between the topology of coverings and the algebra of function fields.
Compact Riemann surfaces are the same as smooth projective complex curves.
So the theorem says that the finite extensions of $\C(z)$ unramified outside $B$ correspond to conjugacy classes of finite-index subgroups of the free group $\pi_1(\hat{\C} \setminus B)$.
It is used to count coverings, as in Hurwitz numbers, and in inverse Galois theory to show that every finite group is a Galois group over $\C(z)$.
Together with the [[uniformization theorem|5EJ7SW1RCV]] it is one of the main existence results for Riemann surfaces.