Quantum Unique Ergodicity for Eisenstein Series on the Hilbert Modular Group over a Totally Real Field

TL;DR

This paper extends quantum unique ergodicity results from classical modular groups to Hilbert modular groups over totally real fields, using Eisenstein series and Hecke operators.

Contribution

It generalizes quantum unique ergodicity to Eisenstein series on Hilbert modular groups over totally real fields, expanding prior results beyond the classical case.

Findings

Proved quantum unique ergodicity for Eisenstein series on Hilbert modular groups.

Provided an expository overview of Hecke operators on non-holomorphic Hilbert modular forms.

Extended Luo and Sarnak's results to a broader class of arithmetic groups.

Abstract

W. Luo and P. Sarnak have proved the quantum unique ergodicity property for Eisenstein series on $\rm{PSL}(2,\mathbb{Z}) \backslash H$. We extend their result to Eisenstein series on $\rm{PSL}(2,O) \backslash H^n$, where $O$ is the ring of integers in a totally real field of degree $n$ over $Q$ with narrow class number one, using the Eisenstein series considered by I. Efrat. We also give an expository treatment of the theory of Hecke operators on non-holomorphic Hilbert modular forms.