Relative Aspects of the Langlands Program, L-Functions and Beyond Endoscopy

ag.algebraic-geometry nt.number-theory rt.representation-theory
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The Langlands program and the theory of automorphic forms are fundamental subjects of modern number theory. Langlands’ principle of functoriality, and the notion of automorphicL-functions, are central pillars of this area. After more than forty years of development, andmany celebrated achievements, large parts of this program are still open, and retain their mystery. The theory of endoscopy, a particular but very important special case of functoriality, has attracted much effort in the past thirty years. This has met with great success, leading to the proof of the Fundamental Lemma by Ngô, and the endoscopic classification of automorphic representations of classical groups by Arthur. Going beyond these remarkable achievements requires new techniques and ideas ; in the past few years, exciting directions have started to emerge, which may renew our vision of the whole subject.

This brings us to the three main topics of this conference : (1) The "relative Langlands program" is a very appealing generalization of the classical Langlands program to certain homogeneous spaces (mainly spherical ones). It relates period integrals of automorphic forms to Langlands functoriality or special values of L-functions.Remarkable progress on the Gan-Gross-Prasad and Ichino-Ikeda conjectures has been made(by Waldspurger, W. Zhang, and others). With the work of Sakellaridis and Venkatesh, this subject has reached a new stage ; we now have a rigorous notion of "relative functoriality",with very promising perspectives. A central tool in all these questions is Jacquet’s relative trace formula, whose reach and theoretical context remain to be fully investigated. (2) Relations between special values of (higher) derivatives of L-functions and height pairings (between special cycles on Shimura varieties and Drinfeld’s Shtuka stacks) will also be part of the program. This includes arithmetic versions of the Gan-Gross-Prasad conjectures, which generalize the celebrated Gross-Zagier formula. In the function field case, striking recent results of Yun and W. Zhang give geometric meaning to higher central derivatives of certain L-functions. (3) Ways of going beyond endoscopy, and proving new cases of functoriality. This includes Langlands’ original idea of using the stable trace formula 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 (including several researchers from Aix-Marseille University) ; to attain a state-of-the-art overview of the different directions that are being actively pursued ; and to promote collaboration and the exchange of ideas between those approaches.


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