On the Kuznetsov Trace Formula for $\mathrm{PGL}_2(\mathbb{C})$

TL;DR

This paper proves the Kuznetsov trace formula for certain discrete groups in PGL_2(C) using a representation theoretic approach, simplifying previous complex analysis methods and leveraging a kernel formula for Bessel functions.

Contribution

It introduces a new proof of the Kuznetsov trace formula for cofinite, non-cocompact groups in PGL_2(C) using representation theory and a kernel formula, avoiding complex analysis.

Findings

Established the Kuznetsov trace formula for PGL_2(C) groups.

Developed a representation theoretic proof method.

Utilized a kernel formula for Bessel functions.

Abstract

In this note, using a representation theoretic method of Cogdell and Piatetski-Shapiro, we prove the Kuznetsov trace formula for an arbitrary discrete group $\Gamma$ in $\mathrm{PGL}_2(\mathbb{C})$ that is cofinite but not cocompact. An essential ingredient is a kernel formula, recently proved by the author, on Bessel functions for $\mathrm{PGL}_2(\mathbb{C})$. This approach avoids the difficult analysis in the existing method due to Bruggeman and Motohashi.