Le programme de Langlands pour les corps de fonctions

Avril - mai 2002
Cours pour étudiants de 2e et 3e cycle 

The Langlands programme for function fields

April - May 2002
Short courses for graduate students

 Organisateurs / Organizers
Henri Darmon (McGill) & Jacques Hurtubise (CRM)
  

 

Horaire / Program

 

 

1ère semaine (8 - 12 avril)

Week 1 (April 8 - 12)

 

 

 

Le mardi 9 avril / Tuesday, April 9

CRM, Pavillon André-Aisenstadt, Université de Montréal, Salle / Room 5340

 

10:00 - 12:00            Jacques Hurtubise (CRM & McGill)

                         «The Hitchin systems I»

 

14:00 - 16:00            Abraham Broer (Montréal)

                         «D-modules I»

 

 

Le jeudi 11 avril / Thursday, April 11

 

10:00 - 12:00            Quebec -Vermont Number Theory Seminar

                     (Burnside Hall, McGill University, Salle / Room 920)

                     M. Spies (Nottingham)

                     TBA

 

14:00 - 16:00            Quebec -Vermont Number Theory Seminar
                      (Concordia University, Library Building, Salle / Room LB-540)

                      G. Shimura (Princeton)

                     "The relative regulator of an algebraic extension"

 

 

2e semaine (15 - 19 avril)

Week 2 (April 15 - 19)

 

 

Le lundi 15 avril / Monday, April 15

CRM, Pavillon André-Aisenstadt, Université de Montréal, Salle / Room 5340

 

10:00 - 12:00            Abraham Broer (Montréal)

                         «D-modules II»

 

 

14:00 - 16:00            Jacques Hurtubise (CRM & McGill)

                         «The Hitchin systems II»

 

 

 

Le mardi 16 avril / Tuesday, April 16

CRM, Pavillon André-Aisenstadt, Université de Montréal, Salle / Room 5340


10:00 - 12:00            Ram Murty (Queen 's)

                         «A survey of the Langlands programme in number fields I»

 

14:00 - 16:00            Amritanshu Prasad (CRM)

                         «Automorphic representations I»

 

 

 

Le mercredi 17 avril/ Wednesday, April 17

CRM, Pavillon André-Aisenstadt, Université de Montréal, Salle / Room 5340

 

10:00 - 12:00            Ram Murty (Queen's)

                         «A survey of the Langlands programme in number fields II»

 

14:00 - 16:00            Amritanshu Prasad (CRM)

                         «Automorphic representations II»

 

 

Le jeudi 18 avril / Thursday, April 18

CRM, Pavillon André-Aisenstadt, Université de Montréal, Salle / Room 5340

 

10:00 - 12:00            Jason Levy (Ottawa)

                         «Trace formulae I»

 

14:00 - 16:00            Jason Levy (Ottawa)

                          «Trace formulae II»

 

 

 

3e semaine (22 - 26 avril)

Week 3 (April 22 - 26)

 

Le mardi 23 avril / Tuesday, April 23

CRM, Pavillon André-Aisenstadt, Université de Montréal, Salle / Room 5340

 

10:00 - 12:00           Ambrus Pal (CRM)

                                  «Introduction to shtukas I»

 

14:00 - 16:00           David Savitt (McGill)

                                  «Étale cohomology I»

 

 

Le mercredi 24 avril/ Wednesday, April 24

CRM, Pavillon André-Aisenstadt, Université de Montréal, Salle / Room 5340

 

 

10:00 - 12:00            Ambrus Pal (CRM)

                                  «Introduction to shtukas II»

 

14:00 - 16:00            David Savitt (McGill)

                                  «Étale cohomology II»

 

 

Le jeudi 25 avril / Thursday, April 25

 

10:00 - 12:00            Quebec -Vermont Number Theory Seminar

                     (Burnside Hall, McGill University, Salle / Room 920)

                      S. Kudla (Maryland)

                     TBA

 

14:00 - 16:00            Quebec -Vermont Number Theory Seminar

                     (Concordia University, Library Building, Salle / Room LB-540)

                     M. Bhargava (Princeton)

                     TBA

 


Résumés / Abstracts

 

Abraham Broer (Montréal)

«D-modules»

 

In these two lectures we will introduce the notion of D-modules on smooth complex algebraic varieties. What are they, examples, constructions, some relations with representation theory. We will discuss versions of the Riemann-Hilbert correspondence and the Decomposition Theorem.

 

Jacques Hurtubise (CRM & McGill)

«The Hitchin systems»

 

A brief introduction to integrable systems will be given, followed by a presentation of the Hitchin integrable systems for arbitrary reductive groups. These are associated to moduli of G-bundles over curves. The quantisation of these systems is an important ingredient of the geometric Langlands programme.

 

Jason Levy (Ottawa)

«Trace formulae»

 

The Selberg-Arthur trace formula gets more complicated as the rank of the group increases, so we will focus on two cases: compact quotient (rank 0) and GL(2) (rank 1). We will produce the trace formula for these groups, and then use them to obtain the Jacquet-Langlands correspondence. We will also give an indication of how the trace formula can be used to obtain automorphic representations from algebraic-geometric objects.

 

 

Ram Murty (Queen's)

«A Survey of the The Langlands Program in number fields»

 

We will outline the basic notions and problems in the Langlands program and attempt to give an update of the status of some the main questions. The main focus will be Artin's conjecture about the holomorphy of non-abelian L-series.

 

 

Ambrus Pal (CRM)

«Introduction to shtukas»

 

Moduli varieties of shtukas, defined originally by V. Drinfeld are the primary objects of study in the course of proof of the Langlands reciprocity law over function fields. I will talk about the basic notions connected to them, such as level structures, Hecke operators. I will also discus their structure in the particular cases of the moduli of shtukas of rank one and two.

 

 

Amritanshu Prasad (CRM)

«Automorphic representations»

 

Automorphic representations, i.e., constituent representations of GL(n,A) occuring in spaces of functions on GL(n,F)\GL(n,A) with a fixed central character are central objects in the Langlands reciprocity. We will discuss the basic analytic and representation theoretic ideas involved in their study, such as Hecke algebras, cuspidal representations, Eisenstein series, and constant terms.

 

David Savitt (McGill)

«Étale cohomology»

 

In this series of introductory lectures, we will define etale cohomology and describe the fundamental results for algebraic varieties, leading up to the Weil conjectures. The talks will be aimed at graduate students unfamiliar with the subject: we will assume a basic familiarity with sheaf cohomology, and with the language of schemes, but little else.

 

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