Raphael Steiner

Fourth moments and sup-norms with the aid of theta functions

It is a classical problem in harmonic analysis to bound $L^p$-norms of eigenfunctions of the Laplacian on (compact) Riemannian manifolds in terms of the eigenvalue. A general sharp result in that direction was given by Hörmander and Sogge. However, in an arithmetic setting, one ought to do better. Indeed, it is a classical result of Iwaniec and Sarnak that exactly that is true for Hecke-Maass forms on arithmetic hyperbolic surfaces. They achieved their result by considering an amplified second moment of Hecke eigenforms. Their technique has since been adapted to numerous other settings. In my talk, I shall explain how to use Shimizu's theta function to express a fourth moment of Hecke eigenforms in geometric terms (second moment of matrix counts). In joint work with Ilya Khayutin and Paul Nelson, we give sharp bounds for said matrix counts and thus a sharp bound on the fourth moment in the weight and level aspect. As a consequence, we improve upon the best known bounds for the sup-norm in these aspects. In particular, we prove a stronger than Weyl-type sub-convexity result.