Duke Mathematical Journal

Large values of eigenfunctions on arithmetic hyperbolic surfaces

Djordje Milićević

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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

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

First available: 27 October 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]


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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