In my notes for a Geometry of Surfaces course that I'm studying, there is the following quote:

(For those of you who like algebra and Galois theory) Studying compact connected Riemann surfaces is in fact equivalent to studying function fields $K(S)$ which are algebraic extensions of $\mathbb{C}$ of transcendence degree 1 (a purely algebraic problem). This $K(S)$ arises as the field of functions of the smooth projective curve corresponding to $S$.

It says to see the Algebraic Curves course for more information, but apart from the fact that we can view algebraic curves as Riemann surfaces of a certain genus (with the degree-genus formula telling us the number), I don't really have much of an idea what this is saying, especially the line about the field of functions of the smooth projective curve. I also have no idea how Galois theory fits into any of this, even though I've just finished a course on it (which is slightly worrying...).

I'm guessing that this is maybe a higher-dimensional analogue of the fact that meromorphic functions on $\mathbb{P}_\mathbb{C}^1$ form a field, so this can be extended to higher genus surfaces, but I'm not overly sure.

Could anybody please enlighten me a bit as to what this is saying? It sounds really interesting and I would love to learn more about this overlap , but I'm not too sure exactly where to start. It would be nice to have an answer with a brief overview in it, and not just some sources, since I'm really meant to be revising for exams, so don't have masses of time at the moment to explore the bits of maths that I would like to!

  • $\begingroup$ I don't know nearly as much about this as I would like to. But it was (at least the impression I got from it) say you have two Riemann Surfaces $X,Y$. Call the meromorphic function field of both $\mathcal{M}(-)$ respectively. Then, for any meromorphic map $f:X\rightarrow Y$, there is the map $f^{*}:\mathcal{M}(Y)\rightarrow \mathcal{M}(X)$ of the function fields. And then something like, if the mapping is injective it is a cover, the index tells you the number of ramification points, or something else like this that I don't actually know anything about. $\endgroup$
    – Eoin
    May 17 '15 at 14:17
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    $\begingroup$ There is a really nice brief overview of Riemann surfaces, algebraic curves and the like in Algebraic Geometry I - Riemann Surfaces and Algebraic Curves by Shokurov edited by Shafarevich. And by brief I mean it covers the main points all the way up to this subject in the first thirty pages (but all four volumes make up 1400 pages...) $\endgroup$
    – Eoin
    May 17 '15 at 14:20

The following three categories are equivalent:

  1. Smooth projective algebraic curves over $\mathbb{C}$ and nonconstant algebraic maps.
  2. Compact connected Riemann surfaces and nonconstant holomorphic maps.
  3. The opposite of the category of finitely generated field extensions of $\mathbb{C}$ of transcendence degree $1$ and morphisms of extensions of $\mathbb{C}$.

Some of the functors between these are easy to describe. $1 \Rightarrow 2$ is given by taking the underlying complex manifold, and $2 \Rightarrow 3$ is given by taking the field of meromorphic functions. But the proofs that these are equivalences is nontrivial (I think $1 \Leftrightarrow 2$ is hard and I don't remember how hard $2 \Leftrightarrow 3$ is).

In particular, studying finite extensions of $\mathbb{C}(t)$ is equivalent to studying branched covers of $\mathbb{CP}^1$ (in either the algebraic or the holomorphic categories), which is how Galois theory fits into all of this.

  • $\begingroup$ Interesting! My knowledge of category theory at the moment is limited to the first few chapters of Allufi's (amazing) book, Algebra: Chapter 0, so the intricacies of this will be lost on me. But this is still great, thanks! Do you have any references for applications of this equivalence? As in, Galois theory being used to solve problems about Riemann surfaces, etc.? $\endgroup$
    – Tim
    May 17 '15 at 18:24
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    $\begingroup$ @Tim: I don't know if this counts as an application, but for xample, an interesting source of Galois extensions is given by taking fixed fields of a finite group $G$ acting on $\mathbb{C}(t)$. These correspond geometrically to certain branched covers $\mathbb{CP}^1 \to \mathbb{CP}^1$. For details see people.reed.edu/~jerry/Quintic/quintic.html. $\endgroup$ May 17 '15 at 18:40
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    $\begingroup$ At a more sophisticated level, this is an entry point to first the topological version of Galois theory, which is covering space theory (en.wikipedia.org/wiki/Covering_space), and second to a generalization of Galois theory in algebraic geometry which has various applications (en.wikipedia.org/wiki/%C3%89tale_fundamental_group). $\endgroup$ May 17 '15 at 18:41
  • $\begingroup$ Would anybody give me a reference to how to prove the fact of (anti)equivalence of the category of compact Riemann surfaces and the category of finite extensions of $\mathbb{C}(z)$? $\endgroup$
    – Vladislav
    Feb 9 '19 at 21:25
  • $\begingroup$ @Vladislav: that's not the equivalence; I was pretty specific about "the category of field extensions of $\mathbb{C}$ of transcendence degree $1$" (otherwise known as "function fields over $\mathbb{C}$") for a reason. The category of finite extensions of $\mathbb{C}(z)$ is the opposite of the category of branched covers of $\mathbb{CP}^1$; morphisms in this category are more restricted than in the full category of compact Riemann surfaces. Anyway, AFAIK this equivalence is quite hard to prove from scratch; you might find a proof in standard texts on Riemann surfaces, e.g. Miranda's. $\endgroup$ Feb 10 '19 at 0:10

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