This project centers around the interactions between periods of automorphic forms, automorphic representations, and arithmetic algebraic geometry. In particular the PIs propose to work on several problems on the general derivative conjecture, analysis and arithmetic of Fourier coefficients on exceptional groups, bounds on heights, computational study of nontempered periods, averages/nonvanishing of derivatives of L-series. Recently the study of periods has yielded new results and proofs about L-functions, breakthroughs towards conjectures about algebraic cycles, and new perspectives on classical questions of representation theory. Combined with other tools, periods have also enhanced our understanding of equidistribution problems and topology on locally symmetric spaces. The PIs are at the forefront of these developments. The proposed framework presents an ambitious plan to work on and formulate conjectures incorporating/connecting the recent groundbreaking works in these areas.
The research topic is central to several areas of mathematics (arithmetic geometry, automorphic representation theory, analytic number theory). A long range goal of the project is to establish a network of scientists working in automorphic representations, number theory, and arithmetic geometry. The PIs envision a group of PhD students and post-docs participating actively in the proposed Research Retreats and Annual Workshops. This group would include the 15 PhD students presently advised by the PIs.
The award funded a project investigating "periods of automorphic forms." The theory of periods of automorphic forms is a subfield of number theory. These periods extract numerical invariants, ``L-functions'', from the extremely complicated mathematical structures called modular forms. These L-functions are amenable to numerical computation, on the one hand; on the other hand, they measure objects that are very difficult to access directly, such as the number of solutions to Diophantine equations. In this way L-functions serve as a bridge between the analytic and the algebraic side of number theory. The simplest example is of such an L-function is the Riemann zeta function, discovered in the 1800s by Riemann in an attempt to better understand the distribution of prime numbers. In broader context, the field of number theory provides some of the mathematical basis for modern cryptography. Although the objects of cryptography are not directly related to automorphic forms, the theory of L-functions nonetheless provides one of the most powerful mathematical tools available for proving rigorous results about these objects. The two core outcomes of the award were as follows: The PI completed a long work with Y. Sakellaridis giving a new and unified framework to study periods of automorphic forms; previously, this study proceeded on a "case by case" basis. The key idea of this work is to use the theory of "spherical varieties" to provide a framework for the theory of periods. Secondly, the PI also discovered a new class of conjectures concerning periods of "cohomological" automorphic forms. This new class of conjectures is of interest because it seems to suggest truly new algebraic structures. The PI will continue this investigation in future work. Seven graduate students participated in the project, and thus the grant supported a significant amount of training for young mathematicians.
