Duke Mathematical Journal

Large values of eigenfunctions on arithmetic hyperbolic surfaces

Djordje Milićević

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Abstract

We prove a new omega result for extreme values of high-energy Hecke-Maass eigenforms on arithmetic hyperbolic surfaces. In particular we show that they exhibit much stronger fluctuations in the $L^{\infty}$-aspect than what the random wave conjecture would have predicted. We adapt the method of resonators and connect values of eigenfunctions to global geometry of these surfaces by employing the pre-trace formula and twists by Hecke correspondences.

Article information

Source
Duke Math. J. Volume 155, Number 2 (2010), 365-401.

Dates
First available: 27 October 2010

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1288185459

Digital Object Identifier
doi:10.1215/00127094-2010-058

Mathematical Reviews number (MathSciNet)
MR2736169

Zentralblatt MATH identifier
1219.11071

Subjects
Primary: 11F37: Forms of half-integer weight; nonholomorphic modular forms
Secondary: 11F32: Modular correspondences, etc. 11N56: Rate of growth of arithmetic functions 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx] 81Q50: Quantum chaos [See also 37Dxx]

Citation

Milićević, Djordje. Large values of eigenfunctions on arithmetic hyperbolic surfaces. Duke Mathematical Journal 155 (2010), no. 2, 365--401. doi:10.1215/00127094-2010-058. http://projecteuclid.org/euclid.dmj/1288185459.


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