Sup-norm and nodal domains of dihedral Maass forms

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 $\varphi$ be a dihedral Maass form with spectral parameter $t_{\varphi }$, then we prove that $\Vert \varphi {\Vert }_{\infty }\ll t_{\varphi }^{3/8+\epsilon }\Vert \varphi {\Vert }_2$, which is an improvement over the bound $t_{\varphi }^{5/12+\epsilon }\Vert \varphi {\Vert }_2$ 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 Lindelöf Hypothesis. Unconditionally, we prove that the number of nodal domains grows faster than $t_{\varphi }^{1/8-\epsilon }$ for any $\epsilon >0$ for almost all dihedral Maass forms.

20 pages. Final version. Referees' comments incorporated, especially for Lemma 14 and its proof. To appear in Comm Math Phys

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