Math 390C: Modular Forms and the Langlands Program (56502)

Course content, format and syllabus

Unfortunately, the Langlands program has a well-deserved reputation for technical difficulty! The aim of this course will be to provide a broad "collquium-level" overview of the main ideas of the Langlands program for non-specialists. It will emphasize broad perspectives on the Langlands program, and its relationship to more concrete questions in number theory and algebraic geometry, over precise proofs and technical detail.

For prerequisites, I'm going to assume a basic understanding of factorization and ramification in number fields, along the lines of the first half of an algebraic number theory course. Of course, we will touch on a wide variety of other topics over the course of the semester, primarily in algebraic geometry and representation theory. Knowledge of these will definitely be helpful in getting more out of the course, but I will make an effort to make the course accessible to those less familiar with these subjects.

Here's a rough overview of the topics I hope to cover.

1. Class Field Theory

   • Global class field theory
   • Local class field theory
   • The Adelic Formulation

2. Tate's Thesis and L-functions

   • Dirichlet L-functions and their functional equations
   • Local L-factors
   • The Global functional equation

3. Admissible Representations

   • Basic definitions
   • Local L-factors
   • The spherical Hecke algebra
   • The unramified local Langlands correspondence
   • Statement of the general correspondence

4. Automorphic Representations

   • Definitions
   • Basic structure theory
   • The global Langlands conjecture for GL_n

5. Modular forms

   • Analytic definition
   • Modular curves
   • Interpretation via algebraic geometry
   • Interpretation via automorphic representations
   • The q-expansion principle
   • Hecke operators

6. The Eichler-Shimura construction

7. Modularity for Elliptic curves (time permitting)

Useful Resources

For Weil groups, the canonical reference is Tate's article "Number-Theoretic Background"; it can be found online here.

For Tate's thesis, aside from the thesis itself, which is a model of readability, one has Kevin Buzzard's lecture notes, which also carefully develop the Fourier theory. There is also the book of Ramakrishnan and Valenza (Springer, GTM 186), which I have not read but which I've seen recommended in several places.

Bushnell-Henniart's The local Langlands conjecture for GL(2) is an outstanding source for many of the important ideas surrounding the local Langlands correspondence, including the basic theory of smooth representations of p-adic groups and a nice discussion of Weil-Deligne representations and their L and epsilon factors. It also gives a complete and explicit description of the local Langlands correspondence for GL(2).

The papers of Bernstein-Zelevinski and Zelevinski develop the fundamental theory of parabolic induction and cuspidal support. The key references are:

• J. Bernstein, A. V. Zelevinsky, Representations of the group GL(n,F), where F is a local non-Archimedean field.
• J. Bernstein, A. V. Zelevinski, Induced representations of reductive p-adic groups. I.
• A .V. Zelevinsky, Induced representations of reductive p-adic groups. II.

A complete classification of cuspidal representations of GL_n can be found in Bushnell-Kutzko, The admissible dual of GL(N) via compact open subgroups.