Algebraic Number Theory (Mathematics GR6657)


Instructor

Gyujin Oh (gyujinoh@math.columbia.edu)

Webpage

https://math.columbia.edu/~gyujinoh/Spring2025.html, and Courseworks.

Time and location

MW 11:40AM-12:55PM, Location 507 Mathematics.

Zoom link.

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 Title Reference
1/22 (W) Basic number theory 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) Class field theory Statements of the local class field theory [ANT, Lecture 19], [ANT2, §2], [CF, VI.4]
2/10 (M) Group cohomology 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) Class field theory 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: Second Inequality [ANT2, §12], [AT, Chapters 7, 14], [CF, VII], [Mil, VII]
3/24 (M) Proof of the global class field theory, II: First Inequality [ANT2, §12], [AT, Chapters 7, 14], [CF, VII], [Mil, VII]
3/26 (W) Proof of the global class field theory, III: Brauer groups [ANT2, §12], [AT, Chapters 7, 14], [CF, VII], [Mil, VII]
3/31 (M) Theory of Complex Multiplication Lattices (i.e. complex elliptic curves) [ANT2, §13], [Cox, §10~11]
4/2 (W) Modular functions for SL_2(Z) [ANT2, §13~14], [Cox, §7, 10~14]
4/7 (M) Modular functions for Γ_0(N) [ANT2, §14~15], [Cox, §10~14]
4/9 (W) First and Second Main Theorems of Complex Multiplication [ANT2, §14~15], [Cox, §10~14]
4/14 (M) Tate's Thesis Fourier analysis over locally compact abelian groups [CF, XV], [Poo]
4/16 (W) Local L-factors and local zeta integrals [CF, XV], [Poo]
4/21 (M) Hecke L-functions and global zeta integrals [CF, XV], [Poo]
4/23 (W) Langlands program for GL(1) Local Langlands correspondence [ANT2, §16]
4/28 (M) Automorphic forms and representations [ANT2, §17]
4/30 (W) Global Langlands correspondence, Galois deformations [ANT2, §18]
5/5 (M) Modularity lifting theorems [ANT2, §19]