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Quantum ergodicity of Eisenstein series for Bianchi groups

Doyon Kim, Youngmin Lee

Abstract

We prove the quantum ergodicity of Eisenstein series on the arithmetic hyperbolic 3-manifold $\operatorname{PSL}_2(\mathcal{O}_F)\backslash \mathbb{H}^3$, where $F$ is an imaginary quadratic field with ring of integers $\mathcal{O}_F$ and class number $h_F\geq 1$. This extends the work of Koyama, who proved the result in the case $h_F=1$, and establishes the first instance of quantum ergodicity of Eisenstein series over number fields with nontrivial class groups.

Quantum ergodicity of Eisenstein series for Bianchi groups

Abstract

We prove the quantum ergodicity of Eisenstein series on the arithmetic hyperbolic 3-manifold , where is an imaginary quadratic field with ring of integers and class number . This extends the work of Koyama, who proved the result in the case , and establishes the first instance of quantum ergodicity of Eisenstein series over number fields with nontrivial class groups.
Paper Structure (11 sections, 16 theorems, 145 equations)

This paper contains 11 sections, 16 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Arithmetic hyperbolic 3-manifolds
  3. Maass cusp forms on $\Gamma\backslash \mathbb{H}^3$
  4. Correspondence between adelic and classical functions
  5. Adelization of Maass forms on $\Gamma\backslash \mathbb{H}^3$
  6. Bounds of $L$-functions
  7. Bounds on Hecke $L$-functions
  8. Bounds on $\operatorname{GL}_2$ automorphic $L$-functions
  9. Quantum ergodicity of Eisenstein series
  10. Contribution of Maass cusp forms
  11. Contribution of incomplete Eisenstein series

Key Result

Theorem 1

Let $F$ be an imaginary quadratic field with class number $h_F>1$, and let $X = \operatorname{PSL}_2(\mathcal{O}_F)\backslash \mathbb \mathbb{H}^3$. For a cusp $\eta$ of $\operatorname{PSL}_2(\mathcal{O}_F)$, let $E_{\eta}(v,s)$ denote the Eisenstein series associated to $\eta$. Then, for any compac where $\mu_{\eta,t}=|E_{\eta}(v,1+it)|^2 dV$.

Theorems & Definitions (30)

  • Theorem 1
  • Theorem 2
  • Definition 2.7
  • Definition 3.2
  • Lemma 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof
  • Lemma 3.8
  • ...and 20 more