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Notes from two lectures at a workshop on Galois Representations, Shimura Varieties, and Automorphic Forms at the Fields Institute (12-16 March 2012). (Lecture 1: Representations of GL_2(R), Lecture 2: Automorphic representations).

Notes from a course on the theory of automorphic forms which I taught at Tel Aviv University during the academic year 2007-2008. (The outline of the course turned out to be too optimistic given the two-hours-per-week and one-semester-plus-epsilon that we had.)

The first goal of this course is to introduce the Langlands conjectures, after covering a lot of background material. Then we will discuss basic methods such as the trace formula and integral representations of L-functions. A rough outline of the topics to be covered:

  1. Algebraic groups: basic structure and forms over a non-algebraically-closed field.
  2. The homogeneous space G(Q)\G(A), properties, connection to the classical picture and arithmetic applications.
  3. Representations of real and p-adic groups, the Satake isomorphism and the Local Langlands Conjectures.
  4. Automorphic representations, Global Langlands Conjectures.
  5. Quaternion algebras, Jacquet-Langlands correspondence, the trace formula.
  6. L-functions, Tate's thesis and other integral representations of L-functions.
I assume some familiarity with algebraic geometry, group cohomology and class field theory. However, all results that we need will be recalled.

COURSE NOTES. Please beware that the notes are only part of what was discussed in lectures, may be fragmentary and will be constantly under revision. Their numbers do not correspond to actual lectures, but rather to logical sections. In any case, use them with caution as they have not been checked carefully for mistakes. I would be very grateful for any corrections, comments or suggestions (from anywhere in the world!).

Approximate table of contents.
Lecture 1: Basic properties of algebraic groups.
Lecture 2: Structure and forms over a non-algebraically closed field.
Lecture 2 1/2: Brauer groups, Galois cohomology.
Lecture 3: Recollection of class field theory.
Lecture 4: The automorphic quotient.
Lecture 5: Basic representation theory of real and p-adic groups
Lecture 6: Automorphic forms and representations.
Lecture 7: The Satake isomorphism and automorphic L-functions.
Lecture 8: The Langlands conjectures and arithmetic.
Lecture 9: Harmonic analysis on homogeneous spaces.
Lecture 10: Parabolic induction and restriction, locally and globally.

Summary of recent changes and known issues in the notes.
Some exercises
More exercises
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