Nodal Domains of Maass Forms I

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 (a)-bounds are applied to prove that locally the number of nodal domains of such a form goes to infinity with its eigenvalue.