TY - JOUR

T1 - Sup-Norm and Nodal Domains of Dihedral Maass Forms

AU - Huang, Bingrong

N1 - Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2019/11/1

Y1 - 2019/11/1

N2 - 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 ϕ be a dihedral Maass form with spectral parameter tϕ, then we prove that ‖ϕ‖∞≪tϕ3/8+ε‖ϕ‖2, which is an improvement over the bound tϕ5/12+ε‖ϕ‖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 tϕ1/8-ε for any ε> 0 for almost all dihedral Maass forms.

AB - 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 ϕ be a dihedral Maass form with spectral parameter tϕ, then we prove that ‖ϕ‖∞≪tϕ3/8+ε‖ϕ‖2, which is an improvement over the bound tϕ5/12+ε‖ϕ‖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 tϕ1/8-ε for any ε> 0 for almost all dihedral Maass forms.

UR - http://www.scopus.com/inward/record.url?scp=85075188054&partnerID=8YFLogxK

U2 - 10.1007/s00220-019-03335-5

DO - 10.1007/s00220-019-03335-5

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AN - SCOPUS:85075188054

SN - 0010-3616

VL - 371

SP - 1261

EP - 1282

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 3

ER -