Quantum ergodicity of Eisenstein series for Bianchi groups

TL;DR

This paper proves quantum ergodicity of Eisenstein series on certain hyperbolic 3-manifolds associated with imaginary quadratic fields, extending previous results to fields with nontrivial class groups.

Contribution

It extends quantum ergodicity results for Eisenstein series to Bianchi groups over number fields with nontrivial class groups, generalizing prior work.

Findings

Proves quantum ergodicity for Eisenstein series on Bianchi groups with class number h_F ≥ 1.

First demonstration of quantum ergodicity for Eisenstein series over number fields with nontrivial class groups.

Generalizes previous results limited to class number one cases.

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.