Arithmetic, L-functions and Automorphic Forms

This award concerns research in analytic number theory. Most questions in analytic number theory are concerned with quantitatively counting arithmetic objects, such as prime numbers and integral solutions to equations. Understanding such questions plays an important role in applications such as cryptography, information theory, physics, economics, and computer science. Powerful analytic tools have proved fruitful in many arithmetic problems, yet there is a natural barrier in these classical methods that limits their applicability. For example, it is not known if there is a prime between every two consecutive squares, even assuming the Riemann Hypothesis. The goal of this project is to develop and exploit new analytic methods to study arithmetic problems on the boundary of classical methods. The educational component of the project will include training and support for undergraduate and graduate students as well as postdoctoral researchers. The barrier often comes from the square root cancellation for individual exponential sums. One way to break the square root barrier is to exploit cancellations in sums of exponential sums over the moduli, which naturally connects analytic number theory and the theory of automorphic forms. The PI will explore the interactions between these two areas by developing new tools in both fields, including various forms of a Kloosterman refinement of the circle method and explicit Kuznetsov trace formulae for congruence subgroups of higher rank groups. The PI will combine these new techniques to study Diophantine problems over thin sets of the integers such as prime numbers and smooth numbers, to count integral points of algebraic varieties beyond hypersurfaces in few variables, and to establish value distributions (e.g. non-vanishing, extreme values and subconvexity) and asymptotic formulae for moments of families of automorphic L-functions of higher rank groups. 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.