CAREER: L-Functions and Subconvexity

This research project is focused on establishing properties of L-functions, which are functions on the complex plane that encode information about various mathematical structures. An example of an L-function is the Riemann zeta function that encodes information about the prime numbers, which today are essential to the way computer data is securely transferred. Other types of L-functions can, for example, help explain the way waves propagate on certain surfaces, a topic of interest in physics. The educational activities of the project include mentoring a postdoctoral researcher, training graduate students in research, introducing undergraduates to research, and immersing high school students in a summer mathematics program that emphasizes the type of thinking used in research. This project will focus on investigating the subconvexity problem, an important and deep question in the theory of L-functions. The subconvexity problem is connected to equidistribution questions and involves obtaining non-trivial upper bounds for an L-function on its critical line. Such bounds are particularly difficult in “conductor-dropping’’ scenarios. The main goal of this project is to establish new subconvexity bounds for L-functions and push existing bounds towards the gold standard (the so-called Weyl bound). The project will consider the symmetric-square L-functions (or close approximations of these), L-functions at “special points” exhibiting conductor dropping, and strong hybrid bounds for L-functions, such as those of Hecke eigenforms twisted by Dirichlet characters. The methodology will include moments of L-functions in families, reciprocity formulae, and automorphic spectral analysis. 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.