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 \({\phi}\) 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 Lindelöf 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.

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Acknowledgements

The author would like to thank Prof. Zeév Rudnick for suggesting thinking about dihedral forms, and for his valuable discussions and constant encouragement. He also wants to thank Professors Gergely Harcos, Peter Humphries, Junehyuk Jung, Djordje Milićević, and Matthew Young for their interest, comments, and suggestions. The author gratefully thanks the referees for the constructive comments and recommendations which definitely helped to improve the readability and quality of the paper, and especially simplifying the proof of Lemma 14.

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Correspondence to Bingrong Huang.

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The work was supported by the European Research Council, under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 320755.

Communicated by J. Marklof

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Huang, B. Sup-Norm and Nodal Domains of Dihedral Maass Forms. Commun. Math. Phys. 371, 1261–1282 (2019). https://doi.org/10.1007/s00220-019-03335-5

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