Sup-norm and nodal domains of dihedral Maass forms

TL;DR

This paper improves bounds on the maximum size and the number of nodal domains of dihedral Maass forms, advancing understanding of their eigenfunction behavior on hyperbolic surfaces.

Contribution

It provides sharper sup-norm bounds and new lower bounds on nodal domain counts for dihedral Maass forms, under both conditional and unconditional assumptions.

Findings

Sup-norm bound improved to t_φ^{3/8+ε}

Lower bound on nodal domains grows faster than t_φ^{1/8-ε} unconditionally

Better lower bounds for nodal domains intersecting fixed geodesics under Lindelöf Hypothesis

Abstract

In this paper, we improve the sup-norm bound and the lower bound of the number of nodal domains for dihedral Maass forms, which are a distinguished sequence of Laplacian eigenfunctions on an arithmetic hyperbolic surface. More specifically, let $\phi$ be a dihedral Maass form with spectral parameter $t_\phi$, then we prove that $\|\phi\|_\infty \ll t_\phi^{3/8+\varepsilon} \|\phi\|_2$, which is an improvement over the bound $t_\phi^{5/12+\varepsilon} \|\phi\|_2$ given by Iwaniec and Sarnak. As a consequence, we get a better lower bound for the number of nodal domains intersecting a fixed geodesic segment under the Lindel\"{o}f Hypothesis. Unconditionally, we prove that the number of nodal domains grows faster than $t_\phi^{1/8-\varepsilon}$ for any $\varepsilon>0$ for almost all dihedral Maass forms.