This collaborative proposal is concerned with developing a new higher rank version of a fundamental identity known classically as the Kuznetsov trace formula, which relates the spectrum of a certain differential operator to the geometry of the space on which the operator acts. The aim is to establish either asymptotics or strong bounds for all the different terms appearing in the formula. A first application is to obtain the symmetry types of certain thin families of L-functions in various higher rank situations. A broad range of further applications are expected. Additionally, the following research problems will be investigated: a search for a new class of Multiple Dirichlet Series will be executed; supercuspidal representations in higher rank will be studied; the Affine Linear Sieve will be combined with bilinear forms methods to exhibit thin orbits containing an infinitude of primes; and finally, effective infinite-volume counting problems will be attacked.
The theory of automorphic forms, representations, and L-functions is a central theme in modern number theory, and has provided links between such diverse areas of mathematics as algebraic geometry, representation theory, probability, combinatorics, and mathematical physics. Thus progress in the understanding of the aforementioned objects often has a significant impact in other fields. For example, cryptographic algorithms which secure wireless communication for the internet and cellular phones often rely heavily on deep properties of prime numbers. The proposal also includes a significant educational and dissemination component in the mentoring of undergraduate, graduate students, and postdocs working in these evolving parts of mathematics, with the hope of bringing traditionally under-represented goups into the field.
The theory of automorphic forms and L-functions has played a central role in the solution of many fundamental problems in number theory. A major goal of this project was to establish the symmetry types (random matrix models coming from physics) of families of higher rank L-functions as conjectured by Katz and Sarnak. In joint work with Kontorovich, the symmetry types for certain GL(3) families were found. This was the fi rst time such symmetry types were obtained for families not coming from rank one groups. Another goal was to obtain a converse theorem for multiple Dirichlet series (L-functions in several complex variables). In the joint work with Diamantis a converse theorem for multiple Dirichlet series was obtained for the first time. All previous converse theorems concerned Dirichlet series in only one complex variable. Going to several complex variables constituted an important advance. In an application, it was shown that certain Shintani zeta functions coming from the theory of prehomogeneous vector spaces are in fact multiple Dirichlet series of Weyl group type. This established a new connection between the theory of prehomogeneous vector spaces and multiple Dirichlet series. A broader impact activity supported by the award involved the three year book project with Joe Hundley. It resulted in a two volume series (Automorphic representations and L-functions for the general linear group, Volumes 1, 2, Cambridge University Press 2011) which provide an elementary exposition of the theory of automorphic representations and L-functions for the general linear group in an adelic setting. The books include concrete examples of global and local representations of GL(n), and present their associated L-functions. Several proofs are given for the first time, including Jacquet's simple and elegant proof of the tensor product theorem.
