This proposal is centered on problems concerning quantitative equidistribution properties of elliptic Hecke eigenforms and its generalization to higher dimensional arithmetic quotients of Hermitian symmetric spaces, as well as the analytic aspects of automorphic forms and their L-functions related to nonvanishing and subconvexity of the L-values, with the aim to gain deeper understanding of them from analytic perspective. The problems proposed lie at the heart of recent research and development in analytic number theory. This proposal emphasizes different approaches to study these problems, and points interesting connections with other subjects.
Automorphic L-function is fundamental object in mathematics. The equidistribution problems proposed to study have origin from quantum mechanics, and is closely related to the theory of automorphic L-function. They are at the interface of number theory, analysis, and geometry. Our approach employs advanced tools from harmonic analysis and number theory, in combination with the new progress from the theory of automorphic representation. It is the rich interplay and interaction of ideas across different fields that we want to pursue in the current proposal.
