CAREER: Endoscopy and Arithmetic in the Relative Langlands Program

The notion of an L-function lies at the core of modern number theory and arithmetic geometry. With origins in the 19th century study of prime numbers by Riemann and Dirichlet, this rich family of functions often serve as a bridge connecting several areas of mathematics including Diophantine equations, geometry, representation theory, and analysis. Examples arise both in the study of solutions to polynomial equations and in the theory of highly symmetric functions known as automorphic forms.

The paradigm of the Langlands program indicates that the rich symmetry properties of automorphic forms may be used to prove properties of arithmetic L-functions, which have a wide array of applications. In the modern relative Langlands program, we now have a conjectural framework to make this relationship precise in terms of a duality between L-functions and certain invariants of automorphic forms known as "periods".

In this context, the relative trace formula (RTF) provides a powerful tool from harmonic analysis to decompose an automorphic period into its "irreducible components or modes" which are often directly related to special values of L-functions. While such a formula is often the only tool available to establish the relationship alluded to above, the theory of the relative trace formula is still incomplete.

In this project, the PI will explore several problems in the relative Langlands program using recently developed tools. In addition to providing opportunities for undergraduate and graduate student projects in this rapidly developing field, this project also supports several educational goals including the organization of seminars and the founding of a graduate student workshop designed to foster interaction among the area's graduate students in automorphic forms and offer opportunities to develop skill in giving professional presentations.

This project will complete the PI's theory of endoscopy in the relative setting and establish the stabilization of a large family of RTFs by developing a theory of twisted relative endoscopy, transfer factors, and variants of Ngo's geometric stabilization in the context of symmetric varieties. Each step will be informed by the connection between relative endoscopy and the duality conjectures of Ben-Zvi, Sakellaridis, and Venkatesh, with a secondary goal of better understanding these conjectures in the arithmetic setting via the geometric aspects of the relative trace formula.

The PI will apply these tools to applications in arithmetic geometry and the relative Langlands conjectures. This includes the case of generalized unitary Friedberg-Jacquet periods, with applications to the arithmetic geometry of certain Shimura varieties. A novel component of this project is to combine these two branches and develop a novel "arithmetic" theory of endoscopy, and to prove new endoscopic arithmetic fundamental lemmas.

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.