Learning seminar on Eisenstein series and the genesis of the Langlands program

The goal of this groupe de travail is to get a better understanding of the mathematics that led Langlands to his conjectures and philosophy. More concretely, we will follow how computing constant terms of Eisenstein series led Langlands to defining automorphic L-functions and formulating functoriality (see Langlands' short note). This will involve learning about automorphic representations and their L-functions, but also looking at the motivating number theory questions coming from class field theory.

The amount of material on these topics and its difficulty is formidable. After a motivational talk, we will try to build up some background. Then, the goals will be to

• sketch how one proves the analytic continuation of the standard L-function;
• have some understanding of the spectral decomposition of Langlands and of the more modern proof of analytic continuation of Bernstein--Lapid;
• do a computation of the constant terms of Eisenstein series that would motivate Langlands' L-Group;
• see some applications.

• Lecture notes on Automorphic L-Functions - Cogdell, Kim, Murty
• An Introduction to Automorphic Representations with a View Towards the Trace Formula - Getz, Hahn
• The Genesis of the Langlands Program - edited by Mueller, Shahidi (pdf only for personal use and in this seminar!)
• An Introduction to the Langlands Program - edited by Bernstein, Gelbart
• Automorphic Forms and Representations - Bump
• Spectral Decomposition and Eisenstein Series - Moeglin, Waldspurger
• Automorphic representations and number theory - Arthur (lecture notes)
• Potentially other more number-theory-oriented writings

Again, the topic can become extremely technical and difficult. The seminar should follow a philosophy of working out examples, motivating and synthesising material. We are also interested in applications and welcome such interludes.

The seminar will take place on Fridays at 14:00 in seminar room 3L15 (note changes below!) at the Institut de Mathématique d'Orsay (exceptions will be announced). The talks will last 2 hours or less, with a 10 minute break in between.

The following plan is subject to change. Part I: We expect to learn about L-functions for GL(n) by the end of December, start of January. Part II: After that we will learn about Eisenstein series, spectral decomposition, and constant terms. Part III: Applications and L-functions coming from number theory. The (preliminary) program is as follows:

• 9/10: Overview, motivation, allocation of talks (Radu Toma)
• 16/10: Tate's thesis I (Lucas Gerth)
• 23/10: Hecke characters with a view towards GL(2) (Drimik Roy Chowdhury)
• 30/10: Holidays! No talk
• 6/11 (salle 3L8!): Modular forms and their L-functions: Lecture I in Cogdell's Lectures on L-functions, Converse Theorems, and Functoriality for GL(n) (pdf) (Radu Toma)
• 13/11 (salle 3L8!): Tate's thesis II (Carsten Peterson)
• 20/11 (salle 3L8!): Automorphic forms: Lecture II in Cogdell's Lectures (Shigenori Nakatsuka)
• 27/11 (need volunteer): Automorphic representations: Lecture III in Cogdell
• 4/12, 11/12: Some local representation theory
• January: Automorphic L-functions for GL(n)

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