Geometric and Functional Analysis

, Volume 21, Issue 6, pp 1375-1418

First online:

Large Values of Eigenfunctions on Arithmetic Hyperbolic 3-Manifolds

  • Djordje MilićevićAffiliated withDepartment of Mathematics, Amherst CollegeMax-Planck-Institut für Mathematik Email author 

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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.

Keywords and phrases

Eigenfunctions quantum chaos method of resonators pre-trace formula Hecke operators 3-manifolds theta correspondence

2010 Mathematics Subject Classification

Primary: 11F37 Secondary: 11F32 11F70 11F72 57M50 58J51 81Q50