Nodal domains of Maass forms II

TL;DR

This paper extends the study of nodal domains of Maass forms to general arithmetic surfaces with reflective symmetry, using topological and representation theoretic methods to derive new restriction theorems and analyze explicit examples.

Contribution

It generalizes previous results to broader classes of surfaces and introduces new restriction theorems leveraging Waldspurger's results, with detailed examples.

Findings

Extended topological arguments to non-compact surfaces with symmetry

Developed new restriction theorems for Maass forms using representation theory

Provided explicit examples illustrating the theoretical results

Abstract

In Part I we gave a polynomial growth lower-bound for the number of nodal domains of a Hecke-Maass cuspform in a compact part of the modular surface, assuming a Lindel\"of hypothesis. That was a consequence of a topological argument and known subconvexity estimates, together with new sharp lower-bound restriction theorems for the Maass forms. This paper deals with the same question for general (compact or not) arithmetic surfaces which have a reflective symmetry. The topological argument is extended and representation theoretic methods are needed for the restriction theorems, together with results of Waldspurger. Various explicit examples are given and studied.