Math 6370-001 (Howe), Fall 2021: Algebraic Number Theory
Advertisement
- The first 2/3 of the class will cover standard introductory material in algebraic number theory. We will emphasize function fields as examples parallel to number fields to illustrate the geometric interpretation of certain concepts.
- The last 1/3 will cover some topics related to Fermat's Last Theorem, irregular primes, modular forms, and the p-adic zeta function, building up to an informal discussion of ideas surrounding Ribet's converse to Herbrand.
Class info
- Spring 2021, Tu/Th 12:25pm-1:45pm (mountain) -- Dicussion/problem sessions
- One recorded lecture + problem sheet will be posted each week. The lecture should be watched before class on Tuesday.
- Office hours by appointment on Zoom or in-person
References and other resources
- Starting week 4 we will use GP/PARI for some computations: PARI/GP
The most important function is nfinit and its variants (bnfinit...) nfinit documentation
- Main source: Milne - Algebraic Number Theory (online course notes)
- Optional: Kato-Kurokawa-Saito - Number Theory 1 (motivational!), Marcus - Number Fields (lots of exercises!), Janusz - Algebraic Number Fields, Cassels and Frohlich - Algebraic Number Theory (advanced), Serre - Local Fields (advanced)
- Towards the end of the course we will discuss some topics that can be found in Serre - A Course in Arithmetic and Washington - Introduction to Cyclotomic Fields.
Recordings and handouts.
Will be posted by Saturday evening the week before.
- Week 15 (12/7, 12/9)
No class on Thursday 12/9.
The final class on Tuesday 12/7 will be an (optional) lecture to tie together the final ideas of the class and suggest some further directions; no problem sheet or recording.
Please fill out the course evaluation when it is emailed to you. Happy holidays!
- Week 14 (11/27, 11/29)
The lecture this week is very short! Not even 20 minutes -- just enough to provide some context as we move into the final topic of the class.
Lecture video. Lecture notes.
Complementary reading: Serre - A Course in Arithmetic, Chapter VII
Week 14 Problem sheet.
- Week 13 (11/23)
No lecture, no notes, no reading -- everything you need is on the problem sheet. The problems work through a first connection between zeta functions and class groups (the analytic class number formula for imaginary quadratic fields), but we won't use this at all in the last two weeks of class, which will study a different connection. The only results necessary from Week 12 are the elementary computations from Week 12 - Exercise 1 (in particular, we don't need anything about the functional equation, Bernoulli numbers, or contour integrals for this week!). Happy Thanksgiving!
Week 13 Problem sheet.
- Week 12 (11/16, 11/18)
No lecture, no notes, no reading -- everything you need is on the problem sheet, which is a complete non-sequitur from what we've done so far.
It'll come back around to class groups next week, but in the meantime you might find yourself digging up a complex analysis book instead of Milne!
Week 12 Problem sheet.
- Week 11 (11/9, 11/11)
Lecture video. Lecture notes.
Complementary reading: Milne - Chapter 7, "Ramification groups" and Chapter 8, "Decomposition groups" through "Applications of the Chebotarev density theorem"
Week 11 Problem sheet.
- Week 10 (11/02, 11/04)
Lecture video. Lecture notes.
Complementary reading: Milne - Chapter 8, "Extending absolute values", "the product formula", and "finiteness theorems"
Week 10 Problem sheet.
- Week 9 (10/26, 10/28)
Lecture video. Lecture notes.
Complementary reading: Milne - Chapter 7, "Extensions of nonarchimedean absolute values" through end of chapter, skipping ramification groups
Week 9 Problem sheet.
- Week 8 (10/19, 10/21)
Lecture video. Lecture notes.
Complementary reading: Milne - Chapter 7, start through "Newton's lemma"
Week 8 Problem sheet.
- Week 7 (10/05, 10/07)
No video or notes this week.
Complementary reading: Milne - Chapter 6 (Cyclotomic Extensions; Fermat's Last Theorem). Note that there are a few things proved in this chapter that appear as guided exercises in the worksheet, so it's better to try the worksheet first then look at this afterwards or if you get stuck.
Week 7 Problem sheet.
- Week 6 (9/28, 9/30)
Lecture video. Lecture notes.
Complementary reading: Milne - Chapter 5 (The Unit Theorem)
Week 6 Problem sheet.
- Week 5 (9/21, 9/23)
Lecture video. Lecture notes.
Complementary reading: Milne - Chapter 3 (Dedekind domains; factorizations) section on "The ideal class group"; Chapter 4 (The Finiteness of Class Number)
Week 5 Problem sheet.
- Week 4 (9/14, 9/16)
Lecture video. Lecture notes.
Complementary reading: Milne - Chapter 3 (Dedekind domains; factorizations) starting at "Factorizations in extensions" through the end
Week 4 Problem sheet.
- Week 3 (9/7, 9/9)
Lecture video. Lecture notes.
Complementary reading: Milne - Chapter 3 (Dedekind domains; factorizations) up through "Integral closures of Dedekind domains"
Week 3 Problem sheet.
- Week 2 (8/31, 9/2)
Lecture video. Lecture notes.
Complementary reading: Milne - Chapter 2 (Rings of Integers)
Week 2 Problem sheet.
- Week 1 (8/24, 8/26)
No recording (in-person introductory lecture on 8/24).
Complementary reading: Milne - Introduction and Chapter 1 (Preliminaries from Commutative Algebra)
Week 1 Problem sheet.