Math 6370: Algebraic Number Theory

Description: In traditional terms, class field theory is the study of abelian extensions of local and global fields. In fact, especially in its modern cohomological formulation, it is much more, serving as a bridge between the classical algebraic number theory of the 19th century (Kronecker-Weber theorem, etc.) and recent developments in the Langlands program (for instance, the proof of Fermat's Last Theorem). This is a course on local and global class field theory, and applications as time permits. It is a sequel to the course with the same name and number offered in spring 2016.

Topics: rapid review of the basics of local and global fields; group and Galois cohomology; local class field theory and local duality; global class field theory and (without proofs) global duality; applications, especially to class groups of cyclotomic fields (Iwasawa theory). That's a lot, but the semester is long.

References: For prerequisites, one could look at Milne's notes on algebraic number theory, the first two chapters of Neukirch's Algebraic Number Theory, or the first two chapters of Cassels-Frohlich, Algebraic Number Theory. For the core course material, references are Milne's notes on class field theory, the other chapters in Cassels-Frohlich (especially those of Serre and Tate), and Cohomology of Number Fields by Neukirch, Schmidt, and Winberg.

Office Hours: Friday 4-5pm (or get in touch with me to arrange another time)

Exercises: I will maintain a running list of problems here. Discuss them with others in the class, and feel free to come talk to me about them as well. Do as many of them as possible, and hand in at least half of the problems from each section (eg, ``Basics of Local Fields") of the problem sheet. (See below for due dates.)

Hand in ``Basics of Local Fields" by September 6.

Hand in ``Basics of Global Fields" by September 20.

Hand in ``Group and Galois Cohomology" by October 4.

Hand in ``Local Class Field Theory" by October 25.

Hand in problems 1-4 of ``Global Class Field Theory" by November 8. Unlike in previous assignments, submit all of the problems.

Hand in problems 5-8 of ``Global Class Field Theory" by November 22. Submit all problems.