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).

Thu Jan 15: Michael Baker, "Global theory of algebraic number fields" (notes shall be edited/expanded soon)

In this talk, I will embark on an informal review of the global theory of algebraic number fields: integrality, discriminants and integral bases, Dedekind domains, fractional ideals and the ideal class group, Minkowski theory, Dirichlet's unit theorem, splitting of primes, as time permits. Motivation for class field theory will be discussed.

Fri Jan 16: Raymond Cheng, "Local theory of algebraic number fields"

Having reviewed the global aspects of algebraic number theory, I will begin to develop the local aspects of the theory. The goal here will be to define and hopefully get to grips with objects known as local fields. To that end, I will speak about absolute values on fields, completions of fields and properties of local fields. Depending on the audience, I may also talk more about the algebraic constructions required for our efforts.

Thu Jan 22: Raymond Cheng, "Local theory of algebraic number fields II"

We will continue our discussion of the local aspects of algebraic number theory. After a brief look at what completions of fields actually look like, we will figure out how absolute values extend in finite field extensions. This will allow us to classify the completions of algebraic number fields. Once all this is done, we will finally discuss local fields and their properties in a little more depth.

Fri Jan 23: cancelled due to Grad Student Colloquium.

Thu Jan 29: Michael Baker, "Group cohomology (and why)"

After a brief recap of the first two weeks of material, I will give a more refined "road map", or big picture, of where we are going. I will then introduce group cohomology, ~~after a hopefully satisfactory explanation of how it will move us closer to our ultimate goals~~.

Fri Jan 30: Ritvik Ramkumar, "Group cohomology II" (this talk and the previous one followed this treatment)

Definition of $G$-modules. The integral group ring $\mathbb{Z}[G]$ and some important ideals. Exact sequences. Free $G$-modules. Hom functors and their exactness properties.

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).

Sat Jan 31: Michael Baker, "Category theory: a framework for reasoning" (somewhat abriged notes courtesy of Ilia Chtcherbakov; video courtesy of Amanda Chan)

Categories, initial and terminal objects, mono and epi morphisms, universal properties, functors, natural transformations, limits, adjunctions, the Yoneda embedding, and monoidal structures.

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.

Thu Feb 5: Ritvik Ramkumar, "Group cohomology and its properties" (switching to Milne's CFT notes as of now; Ritvik will post notes here eventually)

We will start by discussing injective $G$-modules. Then given a $G$-module $M$ we will define the $k$th cohomology group of $G$ with coefficients in $M$ i.e. $H^k(G,M)$ via injective resolutions and discuss some of its functorial properties. We will also give an explicit construction by means of cochains. ~~We will end by computing various examples of cohomology groups. As a consequence we will prove Hilbert's theorem 90~~.

Fri Feb 6: Ritvik Ramkumar, "Tate cohomology and local class field theory"

Continuing from where we left off on Thursday, we will define the crucial Tate cohomology groups and discuss its properties. We will then compute the cohomology of finite cyclic groups. ~~Finally we will prove Tate's theorem. If time permits I will state the main theorems of local class field theory and describe how the theory we developed can be used to prove them~~.

Thu Feb 12: Raymond Cheng, "Tate's theorem and local class field theory" (video)

We shall finally make contact with number theory again. I will begin off by stating the main theorems of local class field theory and by briefly sketching how the theorems will be proved with the tools we have developed. With that in mind, I will review the construction of the Tate cohomology groups and then prove some important properties about them. In particular, I shall discuss Tate's Theorem.

Fri Feb 13: cancelled.

Thu Feb 19: cancelled due to Reading Week.

Fri Feb 20: cancelled due to Reading Week.

Thu Feb 26: Raymond Cheng, "TBA" (video)

TBA (I will move towards establishing the existence of the local Artin map).

Fri Feb 27: cancelled.