student-run seminar on class field theory

Waterloo, Winter 2015

Michael Baker; Raymond Cheng; Ritvik Ramkumar

Meetings: Thursdays (MC 5413) and Fridays (MC 5403), 4:30–6:00 (note new time). Unfortunately, due to lack of time, the seminar is cancelled until further notice.

Talk information will be posted on the Pure Mathematics department page as well as on this page.

The three of us are planning to alternate giving talks on class field theory. Class field theory is a major branch of algebraic number theory that describes the structure of abelian extensions (that is, extensions whose Galois groups are abelian) of global fields (number fields and function fields of curves over finite fields) and local fields. Thanks to class field theory, the abelian case is essentially completely understood, whereas the non-abelian case is still very, very mysterious. Its ideas therefore underlie and inspire much of the current research in number theory; a notable example is the Langlands program, which one can view as providing a kind of conjectural "non-abelian class field theory".

We will be assuming standard background in Galois theory, algebraic number theory, and likely valued fields, etc. So far, it looks like we will be basing the seminar off Neukirch's excellent book Algebraic Number Theory (mostly chapters IV to VI) along with J. S. Milne's excellent notes (see also his notes on algebraic number theory). You may also find these notes (and these) useful if you have never seen valuations before. At least passing familiarity with chapters I and II of Neukirch will probably be necessary (we will begin by reviewing this material), although I am not entirely sure yet whether it will suffice. More information to come soon.

More references (motivation/"big picture"): Class Field Theory Summarized; A Brief Summary of the Statements of Class Field Theory.

Thorough notes will be posted. You can expect them to be posted here each week, usually sometime Friday evening (after both talks have been given).
Note (from Michael). As promised, here is the "preparation" for (both of) the talks Ritvik will give this coming week. I know there are some people who are much less comfortable with this language than others, but it is indispensable for group cohomology: any attempt to avoid it would result in a hopelessly cumbersome exposition that would be traumatic for all parties involved.

I could have easily gone on for an hour about any one of these topics (there's a concept of "comma category" which would have made my exposition of limits a little more slick, but I knew I didn't really have time to go quite that far). If you keep your eyes peeled, you will start seeing examples of these concepts in all the mathematics you learn (not just in the talks next week).
Note (from Ritvik). My talks on Thursday/Friday will use some results from Homological Algebra. While understanding the results at a deep level isn't crucial, it might be helpful to have some idea of what's going on. So I suggest reading Appendix A i.e. pages 86 - 94 of Milne's CFT notes linked below. You should know: The definition of injective objects and injective resolutions. Be comfortable with the statements of the Lemma/Propositions in A.6 - A.10 (Just the statements and how the maps are defined; you can ignore the proofs). Lastly you may assume as a black box that the category of $G$-modules has enough injectives.