MATHEMATICS G6657, Spring 2018: Algebraic number
theory

**IMPORTANT
ANNOUNCEMENT CONCERNING THE FIRST WEEK'S CLASSES**

**NOTE A SECOND CHANGE OF TIME**

There
will be no classes on Wednesday, January 17.

Instead,
there will be a three-hour review of basic
algebraic number theory, and an introduction to adèles, on the morning of
Friday, January 19, from 9-12 am, in room 507.

This year's course will
be devoted primarily to class field theory. Here is a tentative schedule:

1. Adèles and idèles of number fields,
including a proof of finiteness of class number and Dirichlet's unit theorem (2
weeks)

2. Tate's thesis (functional equations of
L-functions of Hecke characters), including the analytic class number formula
(2-3 weeks)

3. Cohomology of groups and Galois
cohomology (2-3 weeks)

4. Global class field theory and
applications (4 weeks)

5. Lubin-Tate formal groups and
local class field theory (2 weeks)

If there is
time, I will also cover the determination of the values at s = 1 of Dirichlet
L-functions and the beginning of the theory of cyclotomic fields.

**REFERENCES (incomplete
list)**

A. Weil, *Basic Number Theory*

J.W.S. Cassels
and A. Fröhlich, *Class Field Theory*

E. Artin and J.
Tate, *Class Field Theory*

S. Lang, *Algebraic Number
Theory*

J.S. Milne, notes on class field
theory

J.-P. Serre, *Local Fields*

D. Cox, *Primes
of the form x ^{2} + ny^{2}*

J. Bernstein,
S. Gelbart, *An Introduction to the
Langlands Program* (especially S. Kudla's chapter on Tate's thesis)

For background
reading, I also recommend Milne's notes on algebraic
number theory, which provide a more complete introduction to algebraic
number theory than my course notes
(but with fewer applications to Diophantine problems).

Grades will be
based on a take home final; undergraduates attending the class will also be
expected to turn in regular homework assignments (roughly every two weeks).

**Return to Michael Harris's home
page**