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

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).$