Algebraic number theory meetings

In the Spring 2016 semester I organized meetings to answer questions and lecture on the background for Prof. Zhang's course on class field theory. We met every Friday 1:30-2:30 in Math 528.


Local Fields: Here are some notes on local fields. I lectured on these during our first two meetings. These notes also contain useful references.

Topics covered/schedule:

Date Topics covered References
1/29 Local fields Local Fields Sections 1-3
[Milne] Ch. 7
2/5 Ramification in local fields
The p-adic exponential and logarithm
Brief introduction to global fields
Local Fields Sections 4-5
[Milne] Ch. 7
2/12 Definition of global field
Rings of integers of number fields
Quadratic fields
[Milne] Ch. 2, or
[Marcus] Ch. 2.
[Lang] Ch. 1 for commutative
algebra background.
2/19 Splitting of primes in number fields
Galois theory and prime decomposition
[Marcus] Ch. 3 and 4
[Milne] Ch. 3
2/26 Local fields from global fields
Examples of studying global fields locally
[Milne] Ch. 7 and 8
3/4 Adeles and ideles
Adelic proof of the finiteness of the class number
[CF] Ch. 2
[Lang] Ch. 7
3/11 Compactness properties of adeles and ideles
Adelic proof of the unit theorem
[CF] Ch. 2
[Lang] Ch. 7
3/18 Spring break
3/25 Minkowski theory
Classical proof of the finiteness of the class number
[Milne] Ch. 4
[Marcus] Ch. 5
4/1 Tate's Thesis preparation
Abstract harmonic analysis
[RV] Ch. 1-3
4/8 Tate's Thesis I
Local functional equation
[RV] Ch. 7
[Lang] Ch. 14
[LCNT] Section 11
4/15 Tate's Thesis II
Global analysis
[RV] Ch. 7
[Lang] Ch. 14
[LCNT] Section 12
4/22 Tate's Thesis III
Global functional equation
[RV] Ch. 7
[Lang] Ch. 14
[LCNT] Section 13
4/29 The Riemann-Roch theorem for number fields
Lecture by Pak-Hin Lee
My paper


[Milne] Milne's notes on Algebraic Number Theory.
A good introduction to the subject.

[Marcus] Marcus, Number Fields.
This book is a nice introduction to, well, number fields. It is very readable, and the last chapter motivates class field theory nicely. The drawback is that the local and adelic theories are nowhere to be found in this book.

[Lang] Lang, Algebraic Number Theory.
Extensive, but perhaps not great for beginners. Also, class field theory is not done using cohomology here. The last part is a nice source for some important analytic aspects of the theory, including Tate's Thesis.

[CF] Cassels and Frohlich, Algebraic Number Theory.
Beautiful. Uses local fields and adeles heavily. The basics are covered very quickly, however. Perhaps it's a better resource for class field theory. This is where original Tate's Thesis was published (though 17 years after it was written.)

[Neukirch] Neukirch, Algebraic Number Theory.
A very extensive and geometric approach to algebraic number theory. It even contains what is essentially the 1-dimensional case of Arakelov Theory. The treatment of class field theory, known as "Abstract Class Field Theory", is due to Neukirch himself. It does not use cohomology.

[Serre] Serre, Local Fields.
It's a theorem, or something, that everything written by Serre is beautiful.

[RV] Ramakrishnan and Valenza, Fourier Analysis on Number Fields.
Very nice and complete introduction to Tate's Thesis, and to the adelic approach to number theory in general. Function fields and number fields are treated on an essentially equal footing here.