Eisenstein series and the Langlands program

Akshay Venkatesh, TTh, 11-12:15, 380-381U.

The goal is to give an introduction to the Langlands program through the theory of Eisenstein series. A detailed investigation of Eisenstein series and their constant terms was, in part, what led Langlands to his conjectures. Thus, to some extent, we will try to follow the same path. We will make use of some clever ideas of Bernstein to simplify the proofs.

It will be useful, although not strictly essential, to have familiarity with:

• holomorphic modular forms, including Hecke operators and L-functions;
• basic functional analysis (e.g., basic properties of compact operators).
• basic theory of reductive algebraic groups (e.g., you should know what a parabolic subgroup is).

Useful references will include:

• Borel, Automorphic forms on SL(2).
• Gelbart, "Introduction to automorphic forms on adele groups."
• Bump's book "Automorphic forms and representations."
• Bernstein/Gelbart (editors): "An introduction to the Langlands program", esp. articles of Kudla and Bump.
• The Corvallis volume.

1. "Review" of classical material.
   • Holomorphic Eisenstein series and rationality of zeta-values.
   • The Euler characteristic of SL_2(Z) is zeta(-1)
   • Real-analytic Eisenstein series; deducing analytic properties of zeta-function. A little bit of scattering theory (Lax-Phillips).
   • Representation theory of SL_2(R), representation-theoretic reformulation of Eisenstein series.
   • Spectral decomposition of L^2(SL_2(R)/Gamma).
2. Adelization and L-functions (reference: Gelbart; we will discuss the first three topics in detail for GL_2 only).
   • Adelization of classical holomorphic forms.
   • Hecke operators and the Satake isomorphism.
   • A little bit of representation theory (mainly as a black box).
   • Adelic theory of L-functions (mainly as a black box).
   • Statement of the local and global Langlands conjectures.
3. Eisenstein series on a general group.
   • Statement of the main theorems.
   • Some of the proofs. (This innocuous-sounding phrase will probably be the hardest and longest part of the course.)
   • Applications: L-functions, Tamagawa numbers, geometry of numbers.
   • The remarkable case of G_2.
4. Other topics, according to time and interest.
   • Arthur conjectures.
   • Basic theory of Shimura varieties.
   • p-adic Eisenstein series and applications.
   • The trace formula.