Analytic problems around automorphic forms and L-functions

One of the main tools for understanding the distribution of prime numbers is through properties of the Riemann zeta function. The zeta function is the most fundamental example of an L-function, which is a mathematical construction that combines arithmetical information about all the primes at once. More general L-functions, such as Dirichlet L-functions, are useful for understanding primes in arithmetic progressions. One of the main ways that L-functions are studied is by placing them into families, such as the family of all Dirichlet L-functions, and viewing their properties through this framework. Much of the proposed work in this proposal concerns the development of properties of new families of L-functions. One of the main goals is to better-understand the size of these L-functions, especially in certain ranges that have been inaccessible using previous methods. The PI will continue to mentor and collaborate with undergraduate students, particularly through the Texas A&M REU. Such opportunities are important for preparing students for graduate studies, particularly for undergraduate students from non-PhD granting institutions as well as from population groups underrepresented in STEM fields. The PI will also continue to advise PhD students to work on problems related to families of L-functions and their moments. The PI will study new families of automorphic forms and their associated L-functions, especially via moments of L-functions and large sieve inequalities. The PI plans to study high moments of L-functions in order to make progress on the challenging but important L-functions in conductor-dropping families. The proposer will also study narrower families of L-functions through the use of new versions of the relative trace formula that isolate small families based on their local behavior. In a related vein, the proposer will study large sieve inequalities for families of automorphic forms, with two main goals. One objective is to establish large sieve bounds in some of the new, narrow families. A second goal is to develop heuristics for conjecturing the size of a large sieve bound for more general families. The PI will mentor PhD students on problems on moments of L-functions for both narrow families and for higher degree L-functions. The proposer will study newform Dedekind sums with his undergraduate students. The methods employed will be techniques from analytic number theory such as functional equations, exponential sums and integrals, and the spectral theory of automorphic forms, including the Arthur-Selberg trace formula and the relative trace formula. 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.