Math 788P is a second course in algebraic number theory, following Michael's introduction in Fall 2012. It is intended to give students an introduction to the language, tools, and power of modern algebraic number theory. As such we will develop much of the "highbrow" algebraic theory.

Frank Thorne, LeConte 400G.

Office hours: Wednesdays, 2:00-5:00. Alternatively, ask me questions after class or find me in the hall.

Comprehensive exam syllabus. This will be added to as the semester progresses.

Course evaluation.

Prerequisites. Michael's algebraic number theory course or equivalent, or a willingness to do extra work on the side. Abstract algebra; students should have taken 703/704 or their equivalent, or be enrolled in them concurrently. (Some material uses Galois theory; this will be taught after Matt covers it in 704.)

Grading. There will be a variety of long homework assignments totalling at least 400 points. The grading scale is A = 200+, B = 140+, giving students flexibility to choose problems matching their background, interests, and curiosity. Students writing their dissertations may be excused from part of the workload (if arranged in advance) at the discretion of the instructor.

There was a midterm exam held on April 1, 2013.

Schedule

Extra classes: To make up for the cancelled classes, we will have an extra class on Tuesday most (but not all) weeks, starting on January 29. The extra classes will be out of the stream of the MWF classes, and will cover a variety of additional topics: adeles and ideles, proof of the Hasse-Minkowski theorem, Galois theory of valuations, the idelic proof of finiteness of the class number, Tate's thesis, Artin L-functions, the Scholz reflection principle, and/or whatever else I decide is cool. These classes will be in seminar style, allowing you to see a variety of interesting advanced topics.

The material covered in the Tuesday classes will not appear on the comprehensive exam.

Homework

(Exception: If you failed to turn in any homework earlier in the semester, then please do turn in a large stack of homework at the end of the semester.)

Homework 1, due January 25.

Homework 2, due February 1.

Homework 3, due February 10.

Homework 4, due February 27.

Homework 5, due March 22.

Homework 5a (concerning valuation theory), due March 22.

Homework 6, due April 19.

Homework 7, due May 8.

Homework 8, due May 8.

For some programs you will want to use Sage or PARI/GP.

Bonus problem (due to Ravi Vakil).

Lecture Notes

(1) The relationship between these notes and what I actually lectured on is a little bit complicated.

(2) There are uncorrected mistakes in these notes. (Please feel free to ask about anything you find suspicious, or otherwise make comments or suggestions.)

Lectures 1-3: Introduction.

Lectures 4-11: Dedekind domains, p-adic numbers, Hasse-Minkowski.

Lectures 12-13.

Lectures 14-21.

Lectures 22-27. (missing 24, to be uploaded)

Lectures 24-33.

Lectures 34-35.

Tuesday Lecture 1: Introduction to valuations in general.

Tuesday Lectures 2-4.

Tuesday Lectures 5-6.

References.

Michael Filaseta's notes.

Rafe Jones, Lecture notes on algebraic number theory. I took Rafe's course as a grad student, it was damn good, and my handwriting back then was much better than it is now. With Rafe's permission, I'll scan and post the notes I took.

J.S. Milne, Algebraic number theory, available free online.

J. Neukirch, Algebraic number theory. Obnxiously expensive, but a masterpiece of a book, for anyone who wants to learn the subject really thoroughly (including class field theory, with complete proofs).

Cassels and Frohlich, Algebraic number theory. An advanced book, a classic, probably the best for learning class field theory. Contains' Tate's thesis.

Murty and Esmonde, Problems in algebraic number theory. I'll probably look here for some homework problems.

Marcus, Number fields. A popular introduction. Not TeXXed, and please don't imitate his notation for the integers mod p, but has a very accessible introduction to Hilbert's ramification theory, and much else besides.

Gouvea, p-adic numbers: an introduction. A lively book, less than forty bucks. Consider buying it.

Serre, A course on arithmetic. Beautiful and tiny. Contains a section on the Hasse-Minkowski theorem which I intend to cover. (You probably don't need to buy it for this course, but you could not go wrong by buying it and reading it)

Lang, Algebraic number theory. Reading Lang is like eating broccoli. Sometimes a bit hard to get down, but extremely good for you.

Cox, Primes of the form x^2 + ny^2. A beautiful book, covering a very interesting bit of algebraic number theory, and also giving an excellent introduction to modular forms. Unfortunately Wiley insists on gouging its customers and is completely unworthy of your business. Ask to borrow my copy instead.

Childress, Class field theory. A more accessible introduction to class field theory than those of Neukirch or Cassels-Frohlich. Very well written.

Topics to be covered:

Note: The list of references to be covered is not exhaustive; most of the references cover all the topics, I have tried to pick the ones which I anticipate relying on the most.

• The p-adic numbers. (4 weeks) Introduced by Kurt Hensel, the p-adic numbers allow us to look ``one prime at a time'' and are ubiquitous in modern algebraic number theory. We will carefully look at their algebraic and topological properties. We will also prove the Hasse-Minkowski theorem, the first example of a local-to-global principle for the solution of Diophantine equations, which will help motivate the study of the $p$-adics.

  References: Gouvea, Neukirch, Milne, Serre.

• The geometry of numbers. (4 weeks.) One way of understanding number fields is to regard their rings of integers as a lattice. We will carry out this construction and learn what insight it gives us. We will study the discriminants and class groups of number fields, and prove classical bounds on these quantities. We will also see (in principle, at least) how to compute the class group of any number field. We might have an entire graduate course on this subject in 2013-2014; if so, then I will abbreviate my treatment.

  References: Neukirch, Marcus.

• The decomposition of prime ideals in extensions. (4 weeks.) A classical topic in algebraic number theory, due to Hilbert, is how prime ideals split in extensions. They can `split', be `inert', or `ramify'. We will learn what this means and see what it has to do with classical questions. For example, a classical result (probably proved in Filaseta's course) gives a concise description of the integers which can be written as a sum of two squares. We will see how this can be interpreted in terms of Hilbert's theory.

  I think Michael covered some of this material in his course, but without the modern language and theory. So, this will build on Michael did. We will develop the (very beautiful) connections with Galois theory and discuss the all-important Frobenius element, and at some point I will mumble the word "Spec", draw a bunch of points on the blackboard, and claim that they are an algebraic curve.

  References: Neukirch, Milne.

• Valuations and their extensions. (depending on time) There is another approach to Hilbert's ramification theory, via the theory of valuations, corresponding to extensions of the $p$-adic numbers. This is the true highbrow approach to algebraic number theory, as it allows us to study number fields globally by looking at their local properties. In other words, think globally, act locally.

  Reference: Neukirch.

• Tate's thesis. (depending on time and interest) we will look at John Tate's visionary 1951 Princeton thesis, which combined algebraic number theory with analytic. Tate started with the $p$-adic numbers and combined all of them to form the ring of adeles, and observed that this was an excellent space on which to do harmonic analysis. This led to a beautiful proof of the functional equation of the Riemann zeta functions (and more generally, of Hecke $L$-functions). Later, it led to a huge variety of generalizations (e.g. the Langlands program), many of them conjectural, and all of them very important and widely studied. We will see what it was that Tate did.

  Reference: Tate (in Cassels-Frohlich).

• Class field theory. We will finish our course with the crowning achievement of algebraic number theory. The basic theorem of class field theory provides an isomorphism between the class group of a number field $K$, and the Galois group of its maximal abelian unramified extension, called the Artin map. We will look at this theorem and its generalizations. The proofs are notoriously difficult, so we will instead focus on understanding the theorems and their applications.

  References: Milne, Cox, Childress. (Neukirch and Cassels-Frohlich for advanced treatments)