, Volume 21, Issue 6, pp 1375-1418
Date: 27 Oct 2011

Large Values of Eigenfunctions on Arithmetic Hyperbolic 3-Manifolds

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Abstract

We prove that, on a distinguished class of arithmetic hyperbolic 3-manifolds, there is a sequence of L 2-normalized high-energy Hecke–Maass eigenforms \({\phi_{j}}\) which achieve values as large as \({\lambda^{1/4+o(1)}_{j}}\) , where \({( \Delta+\lambda_{j} ) \phi_{j} = 0}\) . Arithmetic hyperbolic 3-manifolds on which this exceptional behavior is exhibited are, up to commensurability, precisely those containing immersed totally geodesic surfaces. We adapt the method of resonators and connect values of eigenfunctions to the global geometry of the manifold by employing the pre-trace formula and twists by Hecke correspondences. Automorphic representations corresponding to forms appearing with highest weights in the optimized spectral averages are characterized both in terms of base change lifts and in terms of theta lifts from GSp2.