Nodal domains of Maass forms I

TL;DR

This paper investigates the behavior of nodal domains of Maass forms on the modular surface, establishing bounds on restrictions and showing that the number of nodal domains increases with the eigenvalue.

Contribution

It provides sharp bounds for L^2-restrictions of Maass forms and proves the unbounded growth of nodal domains as eigenvalues increase.

Findings

Sharp upper and lower bounds for L^2-restrictions on curves

Number of nodal domains tends to infinity with eigenvalue

Application of Lindelof Hypothesis and subconvex bounds

Abstract

This paper deals with some questions that have received a lot of attention since they were raised by Hejhal and Rackner in their 1992 numerical computations of Maass forms. We establish sharp upper and lower bounds for the $L^2$-restrictions of these forms to certain curves on the modular surface. These results, together with the Lindelof Hypothesis and known subconvex $L^\infty$-bounds are applied to prove that locally the number of nodal domains of such a form goes to infinity with its eigenvalue.