Topics in automorphic forms
This year the plan is to alternate the course between two interrelated topics, and to emphasize the relations as far as possible.
I. The Arthur-Selberg trace formula and applications.
After a short introduction to the meaning of the Selberg trace formula in the simplest cases, and a few typical applications, this portion of the course will focus on learning the mechanics of the Arthur-Selberg trace formula, including its stable and twisted forms. We will look into deciphering statements of the trace formula in various situations, and show how the statements can be used to prove important cases of Langlands functoriality. One goal is to understand the structure of arguments in Arthur's book The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups.
In the (unlikely) event that time remains, some proofs will also be presented. However, there will be more proofs in the second part of the course, devoted to
II. p-adic L-functions, p-adic periods, and Euler systems.
More precisely, the plan is to study several papers that compute (mostly p-adic) local invariants in cohomology of automorphic Galois representations in terms of p-adic periods, which in turn can be related to integral representations of L-functions. Papers to be studied may include (among others)
The course is provisionally scheduled to meet on Tuesdays and Thursdays from 1:10-2:25, but this may change, and the scheduling will be flexible to accomodate travel schedules of the instructor as well as students. Courses marked in red will be scheduled on January 24 and replace the courses that would normally meet the week of March 6. The first meeting will be January 24.
Tentative weekly schedule
The first part of the course will focus on the local and global Jacquet-Langlands correspondence, both as an application of the trace formula and as the starting point for some striking results of Bertolini-Darmon and Darmon-Rotger.
Week 1, January 24, 26, 27 Introduction to the trace formula
(a) Introduction to the trace formula
(b) Automorphic representations, characters of representations
(c) Automorphic representations and characters, continued
Week 2, January 31, February 2 The Jacquet-Langlands correspondence and applications
(a) Conjugacy classes and the trace formula
(b) The Jacquet-Langlands correspondence, first version
Week 3, February 7, 8, 9 p-adic modular forms
(a) JL correspondence, continued
(b) Automorphic representations attached to classical modular forms
(c) The Shimizu correspondence.
Week 4, February 14, 16 Theta functions, Waldspurger and Gross Formulas
(a) Theta functions, Waldspurger and Gross formulas
(b) Rankin-Selberg L-functions and base change
Week 5, February 21, 23 Selmer groups
(a) Selmer groups, Heegner points, following Bertolini and Darmon
(b) (Guest lecture by Dave Hansen on Coleman integration)
Week 6, February 28, March 2 p-adic L-functions for GL(2)
(a) Jochnowitz congruences, following Bertolini and Darmon
(b) Selmer groups and elliptic curves
No classes March 6-10, spring break March 13-17
Week 7, March 21, 23 Selmer groups, continued
(a) Discussion of Bertolini-Darmon and anti-cyclotomic L-functions
Week 8, March 28, 30 Triple products and cohomology, I
(a) Triple product L-functions and central values
Week 9, April 5 [NOTE CHANGE OF DATE], 6 p-adic triple products
(a) p-adic modular forms and Hida families
(b) p-adic triple products
Week 10, April 11, 12[NOTE CHANGE OF DATE], Abel-Jacobi maps and diagonal cycles
(a) Diagonal cycle classes in de Rham cohomology
(b) Relation with the p-adic triple product
Week 11, April 18, 20, Abel-Jacobi maps in Galois cohomology
(a) Selmer groups of triple products
(b) Explicit reciprocity laws
The schedule for subsequent weeks will be posted as the course proceeds.