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 φ be a dihedral Maass form with spectral parameter t φ , then we prove thatwhich is an improvement over the bound t 5/12+ε φ 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 t 1/8−ε φ for any ε > 0 for almost all dihedral Maass forms. This is to be compared with the "local Weyl law", which gives λ 1/4 φ / log λ φ for any negative curvature surface (see [2]). Iwaniec-Sarnak [21] also proved the same bounds (1.1) for the Date: February 26, 2019.

Bias in the distribution of holonomy on compact hyperbolic 3-manifolds

“…The linear independence hypothesis in Theorem 5.3 is not satisfied when there is an arithmetic progression of spectral parameters, which is known to be true for certain classes of dihedral forms on arithmetic hyperbolic 3-manifolds. Although there are explicit descriptions of dihedral forms on non-compact hyperbolic surfaces (see [Luo13], [Hua19], for example), we were not able to find an explicit construction of dihedral forms on compact hyperbolic 3-manifolds in the literature. Thus, we present a general approach to the construction of dihedral forms on compact arithmetic hyperbolic 3-manifolds (which we refer to as co-compact dihedral forms) and provide an explicit example ] : a i ∈ Z/5Z} which is a field with 5 2 = 25 elements (since p 1 is a maximal ideal).…”

Quantum variance for dihedral Maass forms

“…(see [20,Lemma 11]). These estimates allow us to quickly derive a short Dirichlet polynomial approximation of 1/L(1, φ 2k ) 2 , by a contour integration argument (see [19,Lemma 3] for a similar result).…”