Winter 2013, Section 1

Math 776, as the continuation of Math 676 , is a second-semester graduate course in algebraic number theory. While Math 676 covered a variety of basic topics, Math 776 will focus on a single topic: Class Field Theory, the study of abelian extensions of number fields.

Course details

Time: Mon, Wed, Fri 1:10pm-2:00pm
Place: 2866 East Hall
First meeting: Wednesday, January 9, 2013

About the instructor

Jeffrey Lagarias
Email: lagarias@umich.edu
Web: http://www.math.lsa.umich.edu/~lagarias/m776.html
Office: 3086 East Hall
Office Phone: 763-1186
Office Hours: MWF 2:110--3:00, or by appointment!

Course description

Class field theory, the study of abelian extensions of number fields, was a crowning achievement of number theory in the first half of the 20th century. It brings together, in a unified fashion, the quadratic and higher reciprocity laws of Gauss, Legendre et al, and vastly generalizes them. Some of its consequences (e.g., the Chebotarev density theorem) apply even to nonabelian extensions.

Our approach in this course will be to begin with review and the Kronecker-Weber theorem. Then we cover the formulations of the statements of global class field theory, for number fields, and local class field theory, for p-adic fields.

Textbooks

We will not follow a single text consistently. The primary text will be Serge Lang, Algebraic Number Theory, 2nd Edition Springer-Verlag 1994. This text seems rather dry and unmotivated, but it covers the right things on the analytic side. I hope to supply sufficient motivation in class. A second good source is J.S. Milne's course notes on class field theory, which can be downloaded here. A classical source is G.Janusz Algebraic Number Fields, 2nd Edition (AMS 2005) A very good modern source is Neukirch's Algebraic Number Theory (Springer-Verlag 1999). Students are encouraged to have at least one of these last two sources available.I will point out where the material we cover in class is covered in both.

Other good references:

Cassels and Frohlich, Algebraic Number Theory (excellent, but out of print)
Serre, Local Fields (for local class field theory)
Washington, Introduction to Cyclotomic Fields (for Kronecker-Weber, and some applications)

Prerequisites

Math 676 or equivalent. Graduate coursework in algebra is also recommended. If you are unsure how your background matches these prerequisites, see me.

Homework/Exams

There will be problem sets approximately biweekly for most of the semester. There will be no final exam; instead, students will submit a final paper on a topic not covered during the course. Some spossible topics are listed below (under "Additional Topics"); feel free to come up with others!

Possible Topics

Abelian extensions of the rationals: the Kronecker-Weber theorem (Milne, I.4; Washington, Chapter 14)
Statements of results: classical form (Milne, V.1-V.3)
Some applications of class field theory
Statements of results: modern (adelic) form
Group cohomology (Milne, II)
Local class field theory (Milne, I.1 and III; Neukirch, IV and V)
Proofs of the results of class field theory (Milne, VII; Neukirch, IV and VI)
Additional topics (see below) if time permits.

Additional topics

These are topics not covered in the course, but accessible once we have completed the planned syllabus. In particular, these topics would be suitable for a final paper. This list is subject to change as the semester progresses.

The Lubin-Tate approach to local class field theory (Milne, I.2)
Brauer groups (Milne, IV)
Quadratic forms over number fields (Milne, IV)
The Carlitz module and class field theory for function fields (David Hayes, A brief introduction to Drinfeld modules, in The Arithmetic of Function Fields)
Serre's approach to class field theory for function fields (Serre, Algebraic Groups and Class Fields)
Class field towers: the Golod-Shafarevich inequality (Cassels-Frohlich)
Complex multiplication: explicit class field theory for imaginary quadratic fields (Neukirch, VI.6)
Zeta functions and number fields (Neukirch, VII)
Arakelov divisors, notion of genus for number field (Neukirch, III, see me please)

Jeff Lagarias (lagarias(at)umich.edu)

Homework Assignments:

Homework 0

(due Monday January 14 )

Homework 1

(due Monday January 28)

Homework 2

(due Monday February 18)

Homework 3

(due Wednesday March 13)

Homework 4

(due Monday April 8)