Being far from an expert in the subject I was wondering if people can hint towards a modern exposition of the developments in the last 10 years ? Or if not then suggest some sub-subjects in CFT that are kind of central and popular right now ? Thanks!
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4$\begingroup$ See this answer mathoverflow.net/questions/66500/… $\endgroup$– StoppleCommented Apr 30, 2015 at 19:36
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$\begingroup$ So there is nothing more in the abelian case? $\endgroup$– Dr. PiCommented Apr 30, 2015 at 20:06
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2$\begingroup$ I'm not the right person to write a detailed answer, but there certainly has been lots of work on explicit global class field theory. Relevant links: en.wikipedia.org/wiki/Hilbert%27s_twelfth_problem , en.wikipedia.org/wiki/Stark_conjectures , ams.org/mathscinet-getitem?mr=1884617 $\endgroup$– David E SpeyerCommented Apr 30, 2015 at 20:40
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2 Answers
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I am sufficiently far away from the field that "I keep hearing this a lot" really means something - so probably higher-dimensional class field theory should be on the list of current trends in class field theory. The general goal is to express the abelianization of the étale fundamental group of a variety $X/\mathbb{F}_q$ in terms of some other arithmetic data, such as Chow groups (with modulus). Google search revealed several seminar programs, e.g. here (from Berlin) and here (from Essen). They contain literature references and provide a structured guide to the recent results.
answered May 1, 2015 at 13:38
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$\begingroup$ Also, check out the articles of Alexander Schmidt, mathi.uni-heidelberg.de/~schmidt/publ_de.html $\endgroup$– user19475Commented May 1, 2015 at 14:25
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I believe more work is being done on nonabelian CFT than abelian CFT nowadays (see the link in Stopple's comment), but from the comments I guess you are mainly interested in abelian CFT. I'm no expert in abelian CFT, but my impression is most recent work in CFT proper is studying things like extensions with ramification conditions or explicit constructions. I don't know of any recent surveys, but here are some references you may be interested in:
- Computational class field theory by Cohen and Stevenhagen, a survey of some computational aspects
- Class field theory by Gras, this is a pretty serious book on classical CFT, which covers many recent works as I recall (I don't have a copy on hand), but is already over 10 years old
- You could also browse through some of Franz Lemmermeyer's papers to see some semi-recent work in this area.
