MATHEMATICS GR8659, Spring 2017

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.


Bertolini, Darmon, Rotger:  Beilinson-Flach elements and Euler systems I: Syntomic regulators and p-adic Rankin L-series.


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


Darmon, Rotger:  Diagonal cycles and Euler systems II: The Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin $L$-functions.


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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