Functions of a complex variable Riemann surfaces Theorem

Riemann's existence theorem

Builds on Riemann surface Branched covering Symmetric group Transitive action

Statement

Let $X$ be a compact connected Riemann surface and let $B \subset X$ be a finite set.

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 $p \colon Y \to X$ that extends $p^\circ$ . The pair $(Y, p)$ is unique up to a unique isomorphism over $X$ that is the identity on $Y^\circ$ .
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 $S_d$ with $\sigma_1 \cdots \sigma_n = 1$ that generate a transitive 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. 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 $F(z, w) = 0$ .

Going deeper

For a student

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 it is one of the main existence results for Riemann surfaces.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Tag	`NRJQP7FXFM`
URI	`m2://NRJQP7FXFM`
Kind	statement, theorem
Writing	W0 stub
MSC	30F10, 14H30
Status	This copy of the site has no build data. Run `m2 status NRJQP7FXFM`.
About	Riemann surface, Branched covering
Uses	Riemann surface, Branched covering, Symmetric group, Transitive action
Same as	Wikidata Q134889667 Wikipedia Riemann's existence theorem
Changed	2 November 2026