One obtains more information on covers of a topological space when it carries additional structure, for instance when it is a complex manifold. The complex manifolds of dimension 1 are called Riemann surfaces, and they already have a rich theory. The study of their covers creates a link between the Galois theory of fields and that of covers: finite étale algebras over the field of meromorphic functions on a connected compact Riemann surface correspond up to isomorphism to branched covers of the Riemann surface; by definition, the latter are topological covers outside a discrete exceptional set. As we shall see, all proper holomorphic surjections of Riemann surfaces define branched covers. The dictionary between branched covers and étale algebras over the function field has purely algebraic consequences: as an application, we shall prove that every finite group occurs as the Galois group of a finite Galois extension of the rational function field C(t).

Parts of this chapter were inspired by the expositions in [17] and [23].

Basic concepts

Let X be a Hausdorff topological space. A complex atlas on X is an open covering U = {Ui : i ∈ I} of X together with maps fi : Ui → C mapping Ui homeomorphically onto an open subset of C such that for each pair (i, j) ∈ I2 the map fj o f−1i : fi (Ui ∩ Uj) → C is holomorphic. The maps fi are called complex charts.