Advanced Number Theory - MA843 - Fall 2011

Course Home Page

Course: MA843
Instructor: Jared Weinstein
Lectures: TTh 12:30 pm - 2:00 pm in PRB 146
Office Hours: M 10:00 am - 12:00 pm and Th 2:00 pm - 3:00 pm in MCS 227

In algebraic number theory, one learns about various "reciprocity laws": Quadratic reciprocity, cubic reciprocity, all the way up to the theorems of class field theory. The main theorems of class field theory were mostly established by the 1930s; by then number theorists could give a satisfactory description of all the abelian extensions of a number field K, together with laws predicting how primes of K split in each one.

On the question of developing laws governing the non-abelian extensions of a number field, our best hope currently lies with the theory of automorphic representations, developed by Robert Langlands and many others. Automorphic representations encompass Dirichlet characters (like the Legendre symbol), Hecke characters, and modular forms in one broad sweep. The connection between automorphic representations and number theory is known as the Langlands program. And while humanity remains mired in ignorance when it comes to confirming many of the predictions of the Langlands program, the last few decades have seen some spectacular successes as well. These include the theorem of Langlands-Tunnell, which is an essential piece of the proof of Fermat's Last Theorem, as well as the proofs of the local Langlands conjectures and the Sato-Tate theorem.

This course covers one iota of the theory of automorphic representations. Topics will include Tate's thesis, classical modular forms, adelic automorphic forms, and automorphic representations of adele groups. Our goal is to understand the proof of the Langlands-Tunnell theorem, which states that odd two-dimensional Galois representations with solvable image are modular. (We reserve the right to skip some details.)

Algebraic number theory: Number fields, ideals, splitting of primes in extensions, and completions.

Weekly assignments account for 100% of your grade in this course. Assignments will appear on this page. These are to be turned in to me during lecture, or in my mailbox, or e-mailed to me as a PDF. Working together is encouraged, but everything you turn in must be in your own words.

Assignment 1, due Sept. 15.

Assignment 2, due Sept. 27.

Assignment 3, due Oct. 13.

Assignment 4, due Oct. 25.

Assignment 5, due Nov. 8.

Assignment 6, due Nov. 17

Notes for the lectures on Tate's thesis

An invitation to modular forms

Adelic modular forms

Representations of locally profinite groups

The principal series of GL(2) over a p-adic field

Representations of adele groups

(g,K)-modules

Automorphic Representations

Some files relevent to this course are here.