A Selberg Trace Formula for $\text{GL}_{3}(\mathbb{F}_p)\backslash \text{GL}_{3}(\mathbb{F}_q)/K$

TL;DR

This paper develops a discrete analog of the Selberg Trace Formula for the group GL_3 over finite fields, involving the construction of an upper half space, explicit orbital sum calculations, and spectral decomposition of induced representations.

Contribution

It introduces a novel discrete Selberg Trace Formula for GL_3 over finite fields, including the definition of an analog of the upper half space and explicit analysis of orbital sums and spectral decomposition.

Findings

Established a new trace formula for GL_3 over finite fields.

Explicitly identified double coset spaces and fundamental domains.

Decomposed induced representations to understand spectral contributions.

Abstract

In this paper, we prove a discrete analog of the Selberg Trace Formula for the group $\text{GL}_{3}(\mathbb{F}_q).$ By considering a cubic extension of the finite field $\mathbb{F}_q$, we define an analog of the upper half space and an action of $\text{GL}_{3}(\mathbb{F}_q)$ on it. To compute the orbital sums we explicitly identify the double coset spaces and fundamental domains in our upper half space. To understand the spectral side of the trace formula we decompose the induced representation $\rho = \text{Ind}_{\Gamma}^{G} 1$ for $G= \text{GL}_{3}(\mathbb{F}_q) $ and $ \Gamma = \text{GL}_{3}(\mathbb{F}_p).$