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I'm supervising an undergraduate project on Galois theory and covering spaces. I want to have him read about the fact that from a branched cover of a Riemann surface you get an extension of its field of meromorphic functions, and the Galois groups are the same, but I'm having trouble finding a good book. Fulton's "Algebraic Topology" is OK but rushes through this point. Forster's "Lectures on Riemann Surfaces" looks good but I'd rather not make him learn sheaves. Any recommndations?

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    $\begingroup$ What about "Algebre et théories galoisiennes" of A. Douady and R. Douady (if you can read french)? $\endgroup$ – Henri Feb 17 '12 at 18:00
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    $\begingroup$ McKean and Moll's book Elliptic Curves might be a bit elementary but I like their discussion of this a lot. $\endgroup$ – j.c. Feb 17 '12 at 18:05
  • $\begingroup$ @Henri: that would have been my suggestion, too. $\endgroup$ – Franz Lemmermeyer Feb 17 '12 at 18:06
  • $\begingroup$ Douady and Douady is appealing to me, but it's probably too sophisticated for my student - they define the field of meromorphic functions as a projective limit, for example. McKean and Moll is more the right style, although I'm having trouble finding where they address the fact that I asked about. $\endgroup$ – Nick Addington Feb 18 '12 at 0:09
  • $\begingroup$ Simon Donaldson's new book "Riemann Surfaces" looks very nice if I could scare up a copy... $\endgroup$ – Nick Addington Feb 18 '12 at 0:20

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There is a chapter on Riemann surfaces in Tamās Szamuely's book "Galois Groups and Fundamental Groups", which contains the facts that you are looking for.

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I think V.I.Arnold's lectures "Abel's Theorem in Problems and Solutions" may be a great supplementary reading. The book is basic but beautiful.

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    $\begingroup$ I'm an undergrad who recently started working through this myself. $\endgroup$ – mmm Feb 17 '12 at 22:01
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    $\begingroup$ What a lovely book. Probably too elementary for this student, but I hope I'll find an excuse to use it someday. $\endgroup$ – Nick Addington Feb 17 '12 at 23:09
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    $\begingroup$ There is an apparently different tranaslation of the russina lectures which can be downloaded for free from the website of Sujit Nair. I hope it is allright to point this out here. $\endgroup$ – Michael Bächtold Oct 19 '12 at 8:54
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Here are two sources accesible to an undergraduate

1. Michio Kuga: Galois' Dream. Group theory and differential equations, Birkhauser.

It's written with an undergraduate in mind that is not familiar with the fundamental group and/or covering spaces. He does not cover branched covers though.

2. F. Kirwan: Complex Algebraic Curves, London Math. Soc., Student Texts, vol. 23.

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  • $\begingroup$ Thanks, I will check out the Kuga reference. Kirwan doesn't mention Galois groups at all. $\endgroup$ – Nick Addington Feb 17 '12 at 22:29
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I like a lot:

Algebraic Curves and Riemann Surfaces by Rick Miranda,

published by the AMS. I think it's very suitable for undergraduates.

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    $\begingroup$ This doesn't seem to mention anything the Galois group of a branched cover. $\endgroup$ – Nick Addington Feb 17 '12 at 22:25
  • $\begingroup$ In III.2 it describes cyclic covers of the line, III.3 is about group actions on Riemann surfaces, III.4 treats monodromy. All this stuff seems to me closely related to your question. $\endgroup$ – rita Feb 18 '12 at 10:12
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Try Askold Khovanskii: Galois Theory, Coverings and Riemann Surfaces.

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  • $\begingroup$ A new book by the same author just came out: Topological Galois Theory. $\endgroup$ – Martin Peters Nov 3 '14 at 14:34
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Hi Nick,

Groups as Galois groups by Helmut Völken is a very nice book that I think is suitable for a good undergrad and might have the level you are looking for. I think chapters 4 and 5 are the places where your student should check first, and I think they don't require previous chapters to follow what is there.

Also, Inverse Galois theory by Malle and Matzat is great to see some applications of what he is learning is his project--mainly chapter 1. This one needs more background than the above, so I'm just recommending this one after he has learned the material in the other one.

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I am not sure but the, "introduction to Compact Riemann Surfaces and Dessins d'Enfants" of Ernesto Girondo & Gabino González-Diez. Could be useful

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The first sections of the following two papers contain background material on covering spaces and Galois theory.

Joe Harris Galois groups of enumerative problems Duke Math. J. Volume 46, Number 4 (1979), 685-724.

William Fulton Hurwitz Schemes and Irreducibility of Moduli of Algebraic Curves. The Annals of Mathematics, Second Series, Vol. 90, No. 3 (Nov., 1969), pp. 542-575

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Klaus Lamotke: Riemmansche Flächen, Springer, 2009. http://books.google.pt/books?id=lfrtVbRPtn4C Very strong on algebraic aspects. (Don't know of anything alike in english, yet.)

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Approaching the problem from a slightly different position, you could point your student towards a Masters' thesis:

M. A. D. Robalo, 2009, Galois Theory towards Dessins d’Enfants , Master’s thesis, Instituto Superior Technico, Lisboa.

There are some more or small errors, and the aim is slightly different, so the task might then be to rewrite that (slightly too SGA1 based perhaps), to check for errors, adapting it towards the aims that you have in mind and bringing in more Riemann surface stuff.

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