Large values of eigenfunctions on arithmetic hyperbolic surfaces
type: Item
instance of: scholarly article
title: Large values of eigenfunctions on arithmetic hyperbolic surfaces
zbMATH Open document ID: 1219.11071
DOI: 10.1215/00127094-2010-058
author: Djordje Milićević
published in: Duke Mathematical Journal
publication date: 2011-01-10T00:00:00Z
review text: Let \(K\) be a totally real number field of narrow class number one, and let \({\mathfrak o}=R_k\) be its ring of integers. Let \(A\) be a quaternion algebra over \(K\) ramified at all infinite places except one, let \({\mathcal O}\) be an Eichler \({\mathfrak o}\)-order in \(A\), and let \({\mathcal O}^1\) be the group of elements of norm one. Let \(\rho\) be a \(K\)-embedding of \(A\) into \(M_2({\mathbb R})\), and let \(\Gamma=P\rho({\mathcal O}^1)<\text{PSL}_2({\mathbb R}) \) be the corresponding arithmetic Fuchsian group, acting on the upper half-space \(\mathfrak h\). The Hilbert space \(L_0^2(\Gamma /{\mathfrak h} )=L^2((\Gamma /{\mathfrak h}))\), if \(\Gamma\) is cocompact, \(L_0^2(\Gamma /{\mathfrak h} )=(L^2_{\text{Eis}})^\bot\), if \(\Gamma\) is non-cocompact of finite volume, has an orthonormal basis decomposition \(L_0^2(\Gamma /{\mathfrak h} )=\overline{\oplus_{j\geq 0}{\mathbb C}\phi_j}\), where \( \phi_j\) are Hecke-Maass eigenforms (for \( j\geq j_0\)), with \((\Delta+\lambda_j)\phi_j=0\), \(\Delta\) the Laplacian. The author proves that for every fixed CM-point \(z\in \mathfrak h\), we have, as \(j\to\infty\), \[ |\phi_j(z)|=\Omega\left(\exp\left(\sqrt{\frac{\log\lambda_j}{\log\log\lambda_j}}\left(1+O\left(\frac{\log\log\log \lambda_j}{\log\log\lambda_j}\right)\right)\right)\right). \]
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11F37
Mathematics Subject Classification ID: 11F32
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11N56
Mathematics Subject Classification ID: 58J51
zbMATH DE Number: 5834471
zbMATH Keywords: arithmetic hyperbolic surface
zbMATH Keywords: Hecke-Maass eigenform
zbMATH Keywords: CM-point
zbMATH Keywords: large values
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/00127094-2010-058
OpenAlex ID: W2028417310
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description: scientific article; zbMATH DE number 5834471