\(L^ \infty\)-norms of eigenfunctions for arithmetic hyperbolic 3-manifolds
type: Item
instance of: scholarly article
title: \(L^ \infty\)-norms of eigenfunctions for arithmetic hyperbolic 3-manifolds
zbMATH Open document ID: 0869.11050
DOI: 10.1215/S0012-7094-95-07724-2
author: Shin-ya Koyama
published in: Duke Mathematical Journal
publication date: 1996-03-23T00:00:00Z
review text: Let \(\Gamma\) be an arithmetic subgroup of \(SL_2 (\mathbb{R})\). Then \(\Gamma\) acts on the hyperbolic 2-space \({\mathfrak h}\) with quotient \(X_\Gamma= \Gamma \backslash {\mathfrak h}\) of finite volume. For \(\lambda\geq 0\) let \(\text{Eig} (\lambda)\) denote the \(\lambda \)-eigenspace of the Laplace-Beltrami-operator on \(X_\Gamma\). Let \[ M_\infty (\lambda): =\sup \left\{\left. {|\Phi |_\infty \over|\Phi|_2} \right|\Phi\in \text{Eig} (\lambda) \right\}. \] \textit{H. Iwaniec} and \textit{P. Sarnak} showed in [\(L^\infty\)-norms of eigenfunctions of arithmetic surfaces, Ann. Math., II. Ser. 141, No. 2, 301-320 (1995; Zbl 0833.11019)] that for \(\Gamma\) being the full norm-1-group of a maximal order in a quaternion algebra over \(\mathbb{Q}\) which splits at \(\infty\), we have \[ M_\infty (\lambda)= O\bigl(\lambda^{5/24+ \varepsilon} \sqrt{\text{dim Eig} (\lambda)}\bigr) \] for any \(\varepsilon >0\). Since this result is much sharper than what holds for more general manifolds, it is clear that one will need the arithmetic structure for the proof. In this case a careful analysis of the action of the Hecke-algebra yields the result. In the present paper these methods are transferred to the three dimensional case. For \(\Gamma\) being as above for a quaternion algebra over \(\mathbb{Q}(i)\) one gets: \[ M_\infty(\lambda) =O\bigl(\lambda^{37/76+ \varepsilon} \sqrt{\text{dim Eig} (\lambda)} \bigr). \]
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 35P10
zbMATH DE Number: 784721
zbMATH Keywords: \(L^ \infty\)-norms of eigenfunctions
zbMATH Keywords: arithmetic hyperbolic 3-manifolds
zbMATH Keywords: eigenspace of the Laplace-Beltrami-operator
zbMATH Keywords: hyperbolic 2-space
zbMATH Keywords: norm-1-group
zbMATH Keywords: quaternion algebra
reviewed by: Anton Deitmar
MaRDI profile type: Publication
cites work: Q5820721
cites work: Q5824493
cites work: The spectral function of an elliptic operator
cites work: On the Topography of Maass Waveforms for PSL(2, Z)
cites work: Small eigenvalues of Laplacian for $Γ_{0}(N)$
cites work: \(L^ \infty\) norms of eigenfunctions of arithmetic surfaces
cites work: The behaviour of eigenstates of arithmetic hyperbolic manifolds
cites work: The arithmetic and geometry of some hyperbolic three manifolds
cites work: Q3482231
full work available at URL: s0012-7094-95-07724-2
OpenAlex ID: W1496415642
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description: scientific article; zbMATH DE number 784721