Distribution and Analytic Aspects of Cusp Forms

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This project lies at the interface of number theory, a mathematical discipline concerned with integers, and analysis, the study of continuous phenomena. Just like the familiar light and sound waves, continuous objects such as signals or mass distributions on spaces of varying geometry can be best understood as superpositions of simpler, fundamental harmonics known as eigenfunctions. On curved spaces with a rich set of arithmetic symmetries (arithmetic manifolds), the role of these building blocks closely attuned to their geometry, dynamics, and the underlying algebraic structure is played by cusp forms. This project will investigate the extreme oscillating behavior and geometric impact of non-spherical cusp forms, and the distribution of families of cusp forms within natural ambient spaces. This award will also support graduate students working with the PI.Automorphic forms are basic building blocks of analysis, representation theory, and arithmetic on algebraic groups. From an analytic perspective, cusp forms are joint eigenfunctions of invariant differential operators including the Laplacian, whose long-term/large-scale analytic behavior (such as their size) should reflect the spectral geometry and chaotic dynamics on arithmetic hyperbolic manifolds. The PI will leverage the trace formula, geometry of numbers, and explicit inversion to study such analytic properties of non-spherical cusp forms. Cusp forms naturally occur in families, and it is of central interest to identify the size of a family in expanding and shrinking regions of adelic parameters (generalizing Weyl's law, originally formulated by physicists) and their distribution including symmetry type. In this direction, uniform counting statements and bounds for non-tempered spectrum will be pursued.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.