Sup-Norm and Nodal Domains of Dihedral Maass Forms

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 t_ϕ, 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.