This is probably well-known to the experts or many number theory students, but since I am just starting to learn class field theory (with some basic knowledge of algebraic numbers, e.g. the 3 basic theorems on Dedekind domains, class groups, units; and a little on local/global fields), I would really love to learn what Tate means when he wrote "Higher dimensional class field theory" in the new preface to the Artin-Tate book, as a new direction to the classical class field theory that deals with abelian extensions. So, what does he mean by that? I'd appreciate a summary, and also references/books/papers if necessary.

I can sort of understand that the "usual" class field theory is probably about "1-dimensional objects", made precise by Dedekind's theorem that a prime in a Dedekind domain can be factorized into primes that are determined by a single polynomial (the primes being in the integral closure of the Dedekind domain in a finite separable extension), under certain conditions. Although this is well-known, for definiteness, it's on p. 16 of Professor James Milne's notes on Class Field Theory.

I am not sure how to generalize this to higher dimensions; neither am I sure that the generalization of this theorem answers my own question of what Tate means (and what are "higher dimensional fields" anyway?) I can think of yet another potential meaning of this: from a popular account of what goes into the proof of Fermat's Theorem (the book "Fearless Symmetry"), the quadratic reciprocity law is interpreted as an result on the 1-dimensional Galois representation (I only know the definition of this), so maybe using higher-dimensional Galois representations, maybe one can generalize the class field theory to "higher dimensions". Alas, all these are my speculations and fantasies, probably due to my ignorance on the subject (starting to learn CFT); but I really like to put everything in as nice a perspective (or perspectives) as possible while I am learning CFT. So this leads naturally to my other question below:

How do I find an "optimal" and yet pedagogical (to me) way to learn CFT, incorporating the perspectives and preparations for the most important modern developments in number theory (and potentially algebraic/arithmetic geometry, representation theory, Langlands, etc.)? Are there nice references/books/papers along the way of learning CFT that I should be also looking at?

Thank you very much for your kind suggestions and help.

(EDIT) Thanks to Matt E's explanation of the 2 terms. The 2nd and 3rd to last paragraph of his "Preface to the New Edition" (following link http://books.google.com/books?id=8odbx9-9HBMC&lpg=PP1&dq=class%20field%20theory&pg=PR6#v=onepage&q&f=false) are the relevant passages, where he also wrote down a few other examples of new directions. I only ventured to try to understand the first one, and has a very very rough superficial idea what the 2nd one is generally about - this means I will come back for more of these after I learn more! Since these 2 background passages are a bit long (esp. 3rd to last one, which might give a better context for what Tate means), so please excuse me for providing a link. Please let me know if you can't see the paragraphs from the preview.

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    $\begingroup$ "Higher dimensional class field theory" typically means the class field theory of higher-dimensional local fields, as developed (primarily) by Kato and Parshin. "Non-abelian class field theory" normally means the theory of Galois representations of dimension $> 1$. I'm not sure which Tate is actually referring to, since I don't have a copy of the new preface. Perhaps you could quote the relevant passage in the body of your answer, so that I and others can read it? This would make answering your question easier! $\endgroup$
    – Matt E
    Aug 8 '11 at 3:11

By "higher dimensonal class field theory", Tate means the class field theory of higher dimensional local fields (see also this brief discussion), developed in the work of various people, including Kato and Parshin.

As for your second question, about learning CFT from a modern perspective: with my own students, I encourage them to learn from Cassels and Frolich (including the exercises), from Cox's book Primes of the form $x^2 + n y^2$, and from Washington's article on Galois cohomology in Cornell, Silverman, and Stevens.

The first reference (especially the main articles of Serre and Tate) gives a development of the main results of CFT which I think is hard to beat. Cox's book gives an important classical perspective. Washington's article gives insight into how class field theory can be reformulated as a collection of theorems (mainly due to Tate) on (local and global) Galois cohomology.

Tate's article Number theoretic background in the second volume of Corvalis is good when you have reached a certain level of sophistication, and are ready to move on from just focussing on algebraic number theory and CFT to a broader perspective. My experience is that it is a little austere for a beginner, though.

One thing that you will be missing if you follow the above references is an $L$-function-based perspective on class field theory. I gather that this is discussed in the new edition of Artin--Tate. If so, it is worth learning, since although it is the more old-fashioned point of view on CFT, non-abelian class field theory (i.e. the Langlands program) is founded on the notion of $L$-functions. (I believe that Lang's book also discusses the $L$-function approach to CFT, but I've never read it myself.)

  • $\begingroup$ Thank you very much, this is very informative and useful, I really appreciate it. Best Regards, Poldavian. $\endgroup$
    – Poldavian
    Aug 8 '11 at 4:50

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