Automorphic forms, endoscopy and trace formulas
Formes automorphes, endoscopie et formule des traces

18 – 22 September 2023


Scientific Committee 
Comité scientifique 

Colette Moeglin (Institut de Mathématiques de Jussieu-Paris Rive gauche)
Joachim Schwermer (University of Vienna)
Marie-France Vigneras (Université Paris Diderot)
Jean-Loup Waldspurger (CNRS Institut de Mathématiques de Jussieu-Paris Rive gauche)

Organizing Committee
Comité d’organisation

Raphaël Beuzart-plessis (CNRS, Aix-Marseille Université)
Bertrand Lemaire (CNRS, Aix-Marseille Université)
Marc-Hubert Nicole (Université de Caen)
Louise Nyssen (Université de Montpellier)

The theory of automorphic forms and the Langlands program are fundamental subjects of modern number theory with Langlands’ principle of functoriality and reciprocity being central pillars of this area. However, after more than forty years of intense development, and many celebrated achievements, its most deepest parts still remain elusive.
Since the beginning, the Arthur-Selberg trace formula in its various forms has played a key role in the development of the subject as illustrated e.g. by the seminal work of Jacquet-Langlands. The theory of endoscopy, which has attracted much eorts in the past thirty years and led to many sucess (such as the proof of the fundamental lemma by Ngô or the classication of the automorphic spectrum of classical groups by Arthur), was originally conceived by Langlands-Shelstad as a way to compare two such trace formulas and to develop from such comparisons a new, in many ways more achieved and complete, version of the trace formula called the stable trace formula. It was also used in conjunction with GrothendieckLefschetz trace formulas to study the cohomology of Shimura varieties with applications towards Langlands reciprocity. Both for this and other applications, it was soon realized that we need to extend the original framework to include outer automorphisms of the group under consideration leading to what is now named twisted trace formulas and to adapt the theory of endoscopy to study them.
Although trace formulas has been a very successful tool towards establishing cases of Langlands functoriality/reciprocity, going beyond these remarkable achievements requires new techniques and ideas and in the past few years, exciting directions have started to emerge, which may renew our vision of the whole subject. The main aim of this conference will be to discuss current progress at the forefront of Langlands functoriality and the theory of trace formulas (in all its forms) as well as its applications to a broad range of problems in the Langlands program and beyond. This should in particular include the following topics: 

  • The « relative Langlands program » is a very appealing generalization of the classical Langlands program to include the spectra, both local and automorphic, of homogeneous spaces (mainly spherical ones). In broad terms it relates period integrals of automorphic forms to Langlands functoriality or special values of L-functions with seminal examples provided by the Gan-Gross-Prasad conjectures on which important progress have been made. As for the usual Langlands program, a central tool in all these questions are relative trace formulas, originally introduced by Jacquet and whose reach and theoretical context remain to be fully investigated.  ˆ Generalizations of the functoriality principle, the trace formula and the theory of endoscopy in various contexts such as to covering groups or in the relative setting. 
  • Recent spectacular developments on the front of the Local Langlands Correspondence with the work of Genestier-V. Laorgue for function elds and Fargues-Scholze for number elds will also be included as well as their relation with the trace formula approach and the theory of endoscopy. For function elds, investigations in that direction would probably require to develop a full stabilized twisted trace formula which is not yet available but partial progress have been made on this by Labesse-Lemaire. 
  • Ways of going beyond endoscopy, and proving new cases of functoriality. This includes Langlands’ original idea of using the stable trace formula in a completely new way to study poles of L-functions; but also other related proposals that have attracted lot of recent attention, such as the Braverman-Kazhdan approach through non-standard Poisson summation formulas, or new methods to go « beyond endoscopy in a relative sense, as developed by Sakellaridis.

This conference aims to gather leading experts in this vital area of mathematics; to attain a state-of-the-art overview of the dierent directions that are being actively pursued; and to promote collaboration and the exchange of ideas between those approaches.
This conference will also be the occasion to celebrate the 80th birthday of Jean-Pierre Labesse who has played a prominent role in the development and dissemination of the Langlands program and of the trace formula most specically

description française en gras et ITALIQUE

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to be confirmed