Fourth moments of automorphic forms and an application to diameters of hyperbolic surfaces

Abstract: In joint work with Ilya Khayutin and Paul Nelson, we demonstrate how theta functions may be used to derive geometric expressions for fourth moments of automorphic forms on hyperbolic surfaces. By carefully estimating a second moment matrix count, we obtain a sharp pointwise bound on the fourth moment in the weight and level aspect. As a consequence, we significantly improve the sup-norm bounds in these aspects and give an unconditional upper bound on the diameter of hyperbolic surfaces of the same strength as if one were to assume the Selberg eigenvalue conjecture.

algebraic geometrynumber theory

Audience: researchers in the topic

Séminaire de géométrie arithmétique et motivique (Paris Nord)