Algebraic Number Theory (Mathematics GR6657)
Instructor
Gyujin Oh (gyujinoh@math.columbia.edu)
Time and location
MW 11:40AM-12:55PM, Location 507 Mathematics.
Teaching assistant
Vidhu Adhihetty
Office hours
Tuesdays 1-2PM
Topics
The main objective of this course is to learn about local and global class field theory. We will review necessary backgrounds (e.g. basic algebraic and analytic number theory, homological algebra) and then move on to learn about the statements and the proofs of the local and global class field theory. Time permitting we will also touch upon other related topics (e.g. function fields, elliptic curves with complex multiplication, Langlands program, Iwasawa theory).
Prerequisites
We will assume the knowledge of basic commutative algebra (e.g. GR6261).
References
[ANT] Lecture notes from my course on undergraduate Algebraic Number Theory (GU4043) in Spring 2024.
[ANT2] Current lecture notes.
[AT] Artin--Tate, Class Field Theory.
[CF] Cassels--Frohlich.
[Cox] Cox, Primes of the form x^2+ny^2.
[Mil] Milne, Class Field Theory.
[Poo] Poonen's notes on Tate's thesis.
[Se1] Serre, Local Fields.
[Se2] Serre, Algebraic Groups and Class Fields.
[Tat] Tate, Number theoretic background, in Automorphic Forms, Representations, and L-functions, Part 2 (commonly known as the "Corvallis Proceedings").
Grading
There will be a take-home final exam. Other grading components will be determined after I know how large the class will be.
| Date | Topic | Reference |
|---|---|---|
| 1/22 (W) | Number fields, I | [ANT, Lectures 2~10] |
| 1/27 (M) | Number fields, II | [ANT, Lectures 11~15] |
| 1/29 (W) | Local fields, I | [ANT, Lectures 16~18] |
| 2/3 (M) | Local fields, II | [ANT, Lectures 17~18], [ANT2, §1], [CF, I] |
| 2/5 (W) | Statements of the local class field theory | [ANT, Lecture 19], [ANT2, §2], [CF, VI.4] |
| 2/10 (M) | Cohomology of groups | [ANT2, §3~4], [CF, IV], [Mil, II] |
| 2/12 (W) | Galois cohomology | [ANT2, §3~4], [CF, IV, V, VI.1] |
| 2/17 (M) | Class field theory package | [ANT2, §5], [AT, Chapter 14], [Tat] |
| 2/19 (W) | Adeles | [ANT2, §6], [CF, II] |
| 2/24 (M) | Statements of the global class field theory | [ANT, Lectures 20~21], [ANT2, §7], [CF, VII], [Mil, V] |
| 2/26 (W) | Kronecker--Weber theorems | [ANT2, §8] |
| 3/3 (M) | Proof of the local class field theory | [ANT2, §9], [CF, VI], [Mil, III] |
| 3/5 (W) | Lubin--Tate theory | [ANT2, §10], [CF, VI.3], [Mil, I] |
| 3/10 (M) | L-functions, Analytic class number formula | [ANT2, §11], [CF, VIII] |
| 3/12 (W) | Proof of the global class field theory, I | [ANT2, §12], [AT, Chapters 7, 14], [CF, VII], [Mil, VII] |
| 3/24 (M) | Proof of the global class field theory, II | [ANT2, §12], [AT, Chapters 7, 14], [CF, VII], [Mil, VII] |
| 3/26 (W) | Proof of the global class field theory, III | [ANT2, §12], [AT, Chapters 7, 14], [CF, VII], [Mil, VII] |
| 3/31 (M) | Lattices (= complex elliptic curves) | [ANT2, §13], [Cox, §10-11] |
| 4/2 (W) | The theory of complex multiplication | [ANT2, §13~14], [Cox, §7, 10-14] |
| 4/7 (M) | Explicit class field theory for imaginary quadratic fields | [ANT2, §14], [Cox, §10-14] |
| 4/9 (W) | Tate's Thesis, I | [ANT2, §15], [CF, XV], [Poo] |
| 4/14 (M) | Tate's Thesis, II | [ANT2, §15], [CF, XV], [Poo] |
| 4/16 (W) | Langlands program for GL(1), I: Automorphic side | |
| 4/21 (M) | Langlands program for GL(1), II: Galois side | |
| 4/23 (W) | Langlands program for GL(1), III: Modularity lifting | |
| 4/28 (M) | Langlands program for GL(1), IV | |
| 4/30 (W) | Arithmetic of function fields, I: Analogies | |
| 5/5 (M) | Arithmetic of function fields, II: Global class field theory |