Class Field Theory

Niranjan Ramachandran

This course is a natural sequel to Math 620 (Introduction to Number Theory).

The dominant theme of the course is reciprocity laws (generalizations of Gauss’s Quadratic Reciprocity). In modern terms, this is the study of abelian extensions of number fields (and local fields), i.e., class field theory (CFT); we will begin with the statements of the main theorems of CFT. We will then review the tools of Galois cohomology, Brauer groups, K-theory, formal groups etc before sketching the proofs of the main theorems for both local and global fields. There are two main classical approaches (algebraic and analytic); we will follow the algebraic (cohomological) but needless to say the analytic method is equally important.

The last part of the course (2 weeks) will outline some of the various recent developments in this active field of mathematics (n-local fields, local Langlands correspondence, Wiesend’s theory, the Weil-etale topology, arithmetic duality theorems, geometric CFT, ..)

We shall mainly follow Milne’s notes on CFT. As he states explicitly in the introduction, prerequisites for this course are algebraic number theory (as in Math 620) and graduate algebra (Math 600-601). These prerequisites are amply covered in Milne’s other course notes.

The classical references are

Other sources:

Introductory articles:

- Appendix in Larry Washington’s Cyclotomic Fields,
- Poonen
- Robert (English translation)
- Howe
- Wyman
- Adhikari
- Roquette (various historical papers)
- Ash (Wiles’s theorem as a non-abelian reciprocity law)
- Akhil Mathew (Climbing Mount Bourbaki)

A tentative plan is to have homeworks and a presentation, but this is not yet set. Please feel free to discuss any aspect of the course with me. This course and its organization will be very informal.

(Version of homepage, 29 August 2011)