Quantum Limits of Eisenstein Series in H^3

We study the quantum limits of Eisenstein series off the critical line for \(\mathrm{PSL}_{2}(\mathcal{O}_{K})\backslash\mathbb{H}^{3}\), where \(K\) is an imaginary quadratic field of class number one. This generalises the results of Petridis, Raulf and Risager on \(\mathrm{PSL}_{2}(\mathbb{Z})\bac...

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Bibliographic Details
Published in:arXiv.org 2015-11
Main Author: Laaksonen, Niko
Format: Article
Language:English
Subjects:
Fields (mathematics)
Number theory
Online Access:Get full text
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Summary:We study the quantum limits of Eisenstein series off the critical line for \(\mathrm{PSL}_{2}(\mathcal{O}_{K})\backslash\mathbb{H}^{3}\), where \(K\) is an imaginary quadratic field of class number one. This generalises the results of Petridis, Raulf and Risager on \(\mathrm{PSL}_{2}(\mathbb{Z})\backslash\mathbb{H}^{2}\). We observe that the measures \(\lvert E(p,\sigma_{t}+it)\rvert^{2}d\mu(p)\) become equidistributed only if \(\sigma_{t}\rightarrow 1\) as \(t\rightarrow\infty\). We use these computations to study measures defined in terms of the scattering states, which are shown to converge to the absolutely continuous measure \(E(p,3)d\mu(p)\) under the GRH.
ISSN:2331-8422