Spring 2021, Advanced 2nd year graduate course
Relative aspects of the Langlands program
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This course will be an introduction to automorphic forms and representations, with an emphasis on relative aspects, such as periods and distinction.
- Lecture 1 (March 8, 10h30-12h30)
Low degree L-functions and applications. Notes - Lecture 2 (March 11, 13h-15h)
Hecke characters, abelian reciprocity, Tate's thesis. Notes - Lecture 3 (March 15, 10h30-12h30)
Review of classical modular forms and passage to the group. Notes - Lecture 4 (March 18, 16h30-18h30)
Automorphic forms on real Lie groups and adelization. Notes. - Lecture 5 (March 22, 10h30-12h30)
Real representation theory, mostly examples Notes - Lecture 6 (March 25, 16h30-18h30)
Representation theory of, part I Notes
- Lecture 7 (March 29, 10h30-12h30)
Representation theory of - Lecture 8 (April 1, 16h30-18h30)
The Jacquet--Langlands correspondence, Part I Notes - Lecture 9 (April 6, 10h30-12h30)
The Jacquet--Langlands correspondence, Part II Notes - Lecture 10 (April 9, 15h00-17h00)
The Petersson trace formula: proof by the relative trace formula, and applications. Notes - Lecture 11 (April 12, 10h30-12h30)
The Waldspurger formula (part I) Notes - Lecture 12 (April 15, 15h30-17h00)
The Waldspurger formula (part 2) Notes
The first 2/3 of the class will be an introduction to automorphic forms and representations on GL2, culminating in the statement of local Langlands correspondence and a sketch of proof of the Jacquet-Langlands correspondence. The last 1/3 of the class will focus on relative aspects, mainly through examples: Kuznetsov trace formula, the Waldspurger formula, and distinction of base change lifts. Time permitting, we will dedicate the last lecture to the Langlands functoriality conjectures in a more general setting, or possibly some other related topic.
[Added after the end of the class: we did not have the time to look at distinction of quadratic base change lifts (local or global). Hopefully the local story will be addressed in student oral presentations for the final exam.]
All classes will be given online. Contact me for the Zoom link. Most likely, the exam will be an oral presentation on a topic related to the class.
Here is a preliminary schedule of the lectures (all of which will be in English). More details will follow, along with references. I will update the schedule continually once the class starts.