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\newcommand{\id}{\mathop{\rm id}\nolimits} \)
GALOIS THEORY AND RIEMANN SURFACES
ISHAN LEVY
1. Introduction
The Galois correspondence for fields gives an equivalence between finite extensions and continuous
actions of the absolute Galois group. There is a similar theorem in topology which can be stated as
follows: for every reasonable space, finite covering maps correspond to continuous actions of the
profinite completion of the fundamental group.
These correspondences are highly analogous: the first comes from looking at the maps from the
field extension into a separable closure and acting on them by the Galois group, or by looking at
maps from the universal cover to the space, and acting on them by the fundamental
group.
We will see that in the case of compact Riemann surfaces, this relationship is especially strong.
A compact Riemann surface is determined by its field of meromorphic functions (function field),
which is a transcendence degree \(1\) field extension of \(\CC \), and we will see that Galois theory for the
function field will tell us about finite branched covers of the Riemann surface. This relationship is
clarified after realizing that compact Riemann surfaces all come from some projective
variety.
The first thing we will see is that branched covers are determined by their topological
data.
Theorem 1.1. Given a finite topological branched cover of a compact Riemann surface,
there is a unique holomorphic structure on it making the covering holomorphic.
Proof. After removing the fibres of the branch points this is true because the map is a local
homeomorphism. In order to put holomorphic charts on those fibres, observe that near each
branch point we have a finite cover of a punctured disk, which locally looks like a disk
covering itself via \(z \to z^k\), so we can put back in the branch points using charts that look like \(z \to z^k\).
Given two ways to fill in the branch points, we can make a holomorphic isomorphism from
one cover to the other away from the branch points, which extend to the branch points by
the Riemann extension theorem. □
Any nonconstant map of compact Riemann surfaces is a finite branched cover, and any finite
branched cover as we have seen comes from a unique Riemann surface, so these are the same thing
(i.e we have an equivalence of categories).
Note that we also have that the topological automorphisms of a branched cover are holomorphic.
Indeed, given a topological automorphism, there is both a unique holomorphic structure making
the automorphism a branched cover, and making the cover holomorphic, but the first implies the
second, so by uniqueness, these are the same.
Now we will use a theorem that is obtained from analysis to study maps of compact Riemann
surfaces further.
Theorem 1.2 (Riemann existence theorem). Given \(x_1,\dots , x_n\) on a Riemann surface \(C\), and \(a_1,\dots , a_n \in \PP ^1\), there is
a meromorphic function on \(C\) taking \(x_i\) to \(a_i\).
It is not hard to see that this follows from the fact that any two points can be distinguished by
meromorphic functions.
The meromorphic functions form a field (we don’t count the constant map to infinity as
meromorphic), which we denote \(M(C)\). Given a nonconstant map \(C_1 \to C_2\) of compact Riemann surfaces, we can
pullback meromorphic functions from \(C_2\) to \(C_1\). Since a meromorphic function is determined
locally, and there are points where the map is locally an isomorphism, we get an inclusion
\(M(C_2) \to M(C_1)\).
Theorem 1.3. The degree of the inclusion \(M(C_2) \to M(C_1)\) is the degree of the map \(C_1 \to C_2\).
Proof. First we will show that every meromorphic function on \(C_1\) is the zero of a polynomial
in \(M(C_1)[x]\) of degree \(d\), that of the map \(C_1 \to C_2\).
If the function \(f\) is nonconstant, at every unramified point of \(C_2\), we can take a small
neighborhood, and look at the preimage, which will be \(d\) copies of that neighborhood, each
with a function \(f_1,\dots ,f_d\). We can consider the functions \(e_1,\dots , e_d\) on the neighborhood in \(C_2\) that are the
elementary symmetric functions on the \(f_i\). Since we can glue these together locally, the \(e_i\) define
a meromorphic function on the unramified points of \(C_2\), which extend to all of \(C_2\) by the Riemann
extension theorem. Then \(f\) satisfies \(x^d+\sum e_ix^{d-i} = 0\). To see this, it suffices by the identity principle to observe
it locally at an unramified point, which is true because at an unramified point, the polynomial
factors as \(\prod _1^n(x-a_i)\), where \(a_i\) is a cyclic permutation of \(f\).
Thus by the primitive element theorem, the degree is at most \(d\), as every element is degree
at most \(d\). It suffices to produce an element of degree \(d\). By the Riemann existence theorem, for
an unramified point \(p\), there is a meromorphic function \(f\) taking distinct values on the fibres.
It cannot have degree less than \(d\). Indeed, all the \(a_i\) must be roots of the minimal polynomial
as they all vanish when any one of them does. □
There is a converse:
Theorem 1.4. Given a finite field extension \(K/M(C_2)\), there is a map of Riemann surfaces \(C_1 \to C_2\) such
that the induced field extension is exactly \(K/M(C_2)\).
Proof. Take a primitive element of the field extension, \(f\) with minimal polynomial \(p\). \(f\) is
relatively prime to its derivative, so \(fa+f'b=1\). Away from the poles of the coefficients of \(a,b\), \(f\) and \(f'\) cannot
have common zeros, so have distinct roots. Remove these points and those where \(f' = 0\). Now we
can produce a sheaf on the complement of these points where a section is a function on the
open set satisfying the polynomial relation. This is a locally constant sheaf by the implicit
function theorem, as \(f' \neq 0\), and has \(d\) stalks as the roots are distinct. This gives a holomorphic
cover of the complement of these points, and we can fill in the finitely many points to form
a branched cover of \(C_2\). The degree is clearly \(d\), and moreover there is a tautological function on
it that is a solution to \(p(f)=0\), namely the one that sends the stalk to its value. This is the cover
we want. □
In other words, maps of compact Riemann surfaces are exactly the same as finite field extensions
of the function field!
In particular, we can interpret Galois theory of the function field in terms of branched covers.
Field automorphisms correspond to automorphisms of the branched cover. If we take the quotient
of the Riemann surface by the action of the group of cover automorphisms, the resulting Riemann
surface’s function field is exactly the fixed field of the corresponding automorphism of
fields.
Theorem 1.5. Compact Riemann surfaces are equivalent to finitely generated transcendence
degree \(1\) \(\CC \)-algebras.
Proof. A meromorphic function is a finite map to \(\PP ^1\). By the Riemann existence theorem,
nontrivial meromorphic functions exist, and so the function field is a finite extension of \(M(\PP ^1) = \CC (t)\)
which is exactly a transcendence degree \(1\) extension of \(\CC \). If one is a little more careful, this
gives an equivalence of categories. □
In fact, we can prove more.
Theorem 1.6. Compact Riemann surfaces are the same as smooth projective curves.
Proof. Let \(f\) be a primitive element of \(M(C)\) over \(t\), some finite map to \(\PP ^1\). Then consider the map \(C \to \PP ^2\)
given by \((f:t:1)\). As we saw in Theorem 1.4, we can choose the function \(f\) to distinguish fibres of
the map to \(\PP ^1\) generically, so the generic element of the image of this map has \(1\) preimage. The
image is a singular variety given by the polynomial equation that \(f\) satisfies. By resolving the
singularities of this variety (there are plenty of ways to do this), we can obtain a projective
curve such that if we lift the map to the resolution, we get an isomorphism with our Riemann
surface. In particular, note that the functions \(f,t\) are rational functions on this curve, and every
rational function is meromorphic, so its field of rational functions is exactly that of \(C\). □
Algebraically, the field of rational functions of a variety has all the information of the variety up
to taking dense open sets, which in this case means removing finitely many points. This is why one
should expect the field extensions to correspond to branched covers: there is an algebraic
curve with the same function field such that the branched cover is an actual algebraic
cover.