Endoscopy in the Relative Langlands Program

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The Langlands program of mathematics, at its core, is a vast collection of conjectures which posits that two seemingly distinct areas of mathematics, number theory and representation theory, are connected in deep ways via the study of arithmetically rich functions known as L-functions. These functions arise both in the study of solutions to polynomial equations and in the study of highly-symmetric functions known as automorphic forms. A central insight is that the symmetry properties of automorphic forms may be used to prove properties of L-functions. The so-called relative Langlands program now gives us a conjectural framework describing the relationship between L-functions and certain invariants of automorphic forms known as periods. Analyzing these objects has combined tools from harmonic analysis, algebraic geometry, representation theory, and mathematical physics. In this project, the PI will develop and generalize a novel theory, known as endoscopy, to further study these period integrals of automorphic forms. The project will also support the PI as he mentors graduate and undergraduate students, organizes conferences and workshops, and contributes to the Automorphic Project. The relative trace formula is a central tool in the relative Langlands program for studying automorphic periods. Despite this, its theory has not yet been developed to meet the needs of the broader theory, and it has largely been used in contexts with strong multiplicity-one properties. This project will develop a novel theory of endoscopy and stabilization for relative trace formulae. This can be seen as a generalization of the Langlands-Shelstad-Kottwitz program of stabilization of the Arthur-Selberg trace formula, and will open the technique of comparison of trace formulas to periods where local multiplicity one may fail. The main goals of the project are: (1) to pre-stabilize the elliptic part of the relative trace formula associated to a large family of periods, (2) to develop a robust local and global theory of endoscopy in this setting and consider the stabilization of the relative analogues of the Hitchin fibration following the work of Ngo, (3) establish the central harmonic-analytic conjectures to affect new global comparisons of relative trace formulae in several cases of interest to arithmetic applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.