Large Values of Eigenfunctions on Arithmetic Hyperbolic 3-Manifolds
Large Values of Eigenfunctions on Arithmetic Hyperbolic 3-Manifolds
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算术双曲 3 流形上特征函数的大值
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发表时间：
2011
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通讯作者：
Djordje Milićević
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作者：
Djordje Milićević
证明了在一类特殊的算术双曲三维流形上，存在一个L2归一化高能Hecke-Maass特征形式序列$${Phi_{j}}$$，它的值可以达到$${lambda^{1/4+o(1)}_{j}}$$，其中$${(Delta+lambda_{j})Phi_{j}=0}$$。在算术双曲3-流形上表现出这种特殊行为的流形，在公度上，正是那些包含完全浸没的测地曲面的流形。我们采用谐振子方法，利用预迹公式和Hecke对应的扭转将本征函数值与流形的整体几何联系起来。与优化谱平均中出现的权重最高的形式相对应的自同构表示在基变化升力和来自GSP2的theta升力方面都被表征。
We prove that, on a distinguished class of arithmetic hyperbolic 3-manifolds, there is a sequence of L2-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.