The classic reference of this topic is Serre's Algebraic Groups and Class Fields. However, many parts of this book use Weil's language, which I find quite hard to follow. Is there another reference to the topic, using a more modern language (schemes etc.)?
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5 Answers
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Have you looked at
Ben-Zvi's notes (i seem to remember a video, but i could not find it.) also see his other lectures and videos
Thesis of Peter Toth (follows Deligne's approach)
Kerz's articles (amazing innovations in class field theory by Kerz and Wiesend); there is also an excellent Seminaire Bourbaki expose on this by Szamuely (in French).
There are many other good references, but hope this can help.
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$\begingroup$ I just saw Elencwajg's reply; sorry for the duplicates in my reply. @Elenvwajg: glad to find myself often in your good company! $\endgroup$– SGPCommented Aug 17, 2011 at 15:08
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$\begingroup$ Many many thanks for the very good recommendations. $\endgroup$– QcHCommented Aug 17, 2011 at 23:35
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1) Our (slightly pseudonymous!) friend, Brian Conrad, has written this beautiful introduction to geometric class field theory in his characteristically lucid style.
2) Another friend, Péter Tóth, has just written a Master Thesis "Geometric Abelian Class Field Theory" which seems to be what you are looking for: it is geometric and contains all necessary prerequisites. And the author writes in his abstract that he wants "...to remedy the unfortunate situation that the literature on this topic is very deficient, partial and sketchy written...".
3) David Ben-Zvi, a well-known specialist and another friend of ours, gave talks on Geometric Langlands at MSRI in 2002, and part I is on geometric class field theory. Here is a link
I am very happy and proud that all the specialists mentioned above are members of and active contributors to our site.
answered Aug 17, 2011 at 14:48
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1$\begingroup$ These are very valuable. Thanks a lot! $\endgroup$– QcHCommented Aug 17, 2011 at 23:34
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1$\begingroup$ Also notice Avichai Tendler's Master's Thesis at the Hebrew University of Jerusalem, and several other links that are returned by a web search with the terms "geometric abelian class field theory". $\endgroup$– Alex M.Commented Dec 18, 2016 at 19:31
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I have seen Section e) of the letter from Deligne to Serre available at http://www.math.uni-bonn.de/people/richarz/DeligneAnSerreFeb74.pdf mentioned as a reference. I have not read it myself and it is hand written (in French), but it might be what you are looking for.
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1$\begingroup$ Thanks for the recommendation. Unfortunately, I don't know any French, which makes reading handwriting pretty much impossible. $\endgroup$– QcHCommented Aug 17, 2011 at 14:13
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$\begingroup$ Can’t find anything following the link? $\endgroup$ Commented May 25, 2018 at 19:38
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There is also Milne's Arithmetic Duality Theorems, http://jmilne.org/math/Books/ADTnot.pdf p. 126ff., Appendix I.A. He proves local and global class field theory using algebraic geometry and Galois cohomology.
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In the appendix to chapter I of Pursuing stacks, Grothendieck writes:
A large part of the letter outlines (very sketchily) some main points of a duality program (including a cohomological formulation of “geometric” local and global class field theory), which emerged by the end of the fifties and appears here for the first time in print.
For example App 12 is entitled "Global “geometric” class field theory as a cohomological duality formula. Serre duality and the “Lang trick”." and 13 "Case of local “geometric” class field theory."
Hope this helps!
