\(L^ \infty\) norms of eigenfunctions of arithmetic surfaces
type: Item
instance of: scholarly article
title: \(L^ \infty\) norms of eigenfunctions of arithmetic surfaces
zbMATH Open document ID: 0833.11019
DOI: 10.2307/2118522
published in: Annals of Mathematics. Second Series
publication date: 1996-03-17T00:00:00Z
review text: Let \(H= H^2\) denote the Poincaré upper half plane with Laplace operator \(\Delta= y^2 ({{\partial^2} \over {\partial x^2}}+ {{\partial^2} \over {\partial y^2}})\). Consider a discrete subgroup \(\Gamma\) of \(\text{SL}(2,\mathbb R)\) coming from a quaternion division algebra over \(\mathbb Q\). See \textit{M. Eichler} [Lectures on modular correspondences, Tata Inst. (1957); \url{http://www.math.tifr.res.in/~publ/ln/tifr09.pdf}], \textit{S. Katok} [Fuchsian groups. Chicago: The University of Chicago Press (1992; Zbl 0753.30001)], or \textit{M.-F. Vignéras}, Arithmétique des algèbres de quaternions. Berlin etc.: Springer-Verlag (1980; Zbl 0422.12008)]. Then \(X= \Gamma\setminus H\) is a compact hyperbolic surface. Consider an orthonormal basis \(\varphi_j\) of simultaneous eigenfunctions of \(\Delta\) and the Hecke operators \(T_n\) on \(L^2 (X)\). Let \(\Delta \varphi_j=- \lambda_j \varphi_j\), \(T_n \varphi_j= \lambda_j (n) \varphi_j\). The main theorem of the paper being reviewed says that for all \(j\) and \(\varepsilon>0\), we have \[ |\varphi_j |_\infty \ll_\varepsilon \lambda_j^{(5/ 24)+ \varepsilon} \qquad \text{and} \qquad |\varphi_j |_\infty\gg c\sqrt {\log \log \lambda_j} \] for infinitely many \(j\), where \(c\) is a positive constant. The upper bound beats the more general result of \textit{A. Seeger} and \textit{C. Sogge} [Indiana Univ. Math. J. 38, 669--682 (1989; Zbl 0703.35133)]\ in which \(5/24\) is replaced by \(1/4\). That the \(\varphi_j\)'s are not uniformly bounded is consistent with the numerical experiments of \textit{D. Hejhal} and \textit{B. Rackner} [Exp. Math. 1, 275--305 (1992; Zbl 0813.11035)]. See also \textit{P. Sarnak} [The Schur lectures (1992), Isr. Math. Conf. Proc. 8, 183--236 (1995; Zbl 0831.58045)]. This is also consistent with the basic conjecture (an analogue of the Ramanujan conjecture) that \[ |\varphi_j |_\infty \ll_\varepsilon \lambda_j^\varepsilon. \] Sarnack [loc. cit.]\ points out that upper bounds for \(|\varphi_j |_…
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 35P10
Mathematics Subject Classification ID: 30F99
zbMATH DE Number: 750875
zbMATH Keywords: compact fundamental domain
zbMATH Keywords: Maass wave form
zbMATH Keywords: spectral expansion of an automorphic kernel
zbMATH Keywords: Laplacian
zbMATH Keywords: quaternion division algebra
zbMATH Keywords: eigenfunctions
zbMATH Keywords: Hecke operators
zbMATH Keywords: Ramanujan conjecture
zbMATH Keywords: Selberg trace formula
zbMATH Keywords: Maass cusp forms
Wikidata QID: Q110907561
author: Peter Sarnak
author: H. Iwaniec
reviewed by: Audrey A. Terras
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2118522
OpenAlex ID: W2031572484
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description: scientific article; zbMATH DE number 750875