**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)

Bertolini,
Darmon: A rigid analytic
Gross-Zagier formula and arithmetic applications.

Bertolini, Darmon: Euler systems and Jochnowitz
congruences.

Darmon,
Rotger: Diagonal cycles and Euler
systems I: A p-adic Gross-Zagier formula.

Lei,
Loeffler, Zerbes: Euler systems for Rankin-Selberg convolutions of modular
forms.

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 **

**(under construction)**

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

(b) Continued

Week 8, March 28, 30 Triple products and cohomology, I

(a) Triple product L-functions and central values

(b) Continued

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.

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