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 x2 + ny2
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).
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