The Selberg Trace Formula for Hecke operators on cocompact Kleinian groups

TL;DR

This paper derives the Selberg trace formula for Hecke operators on cocompact Kleinian groups with finite-dimensional representations, providing new insights into eigenvalue distribution and an analogue of Huber's theorem.

Contribution

It extends the Selberg trace formula to Hecke operators in the setting of cocompact Kleinian groups with finite-dimensional unitary representations, offering new applications.

Findings

Derived the trace formula for Hecke operators in this context

Provided results on the distribution of Hecke eigenvalues

Established an analogue of Huber's theorem

Abstract

We compute the Selberg trace formula for Hecke operators (also called the trace formula for modular correspondences) in the context of cocompact Kleinian groups with finite-dimentional unitary representations. We give some applications to the distribution of Hecke eigenvalues, and give an analogue of Huber's theorem.