A Selberg Trace Formula for GL3(Fp)∖GL3(Fq)/K

Daksh Aggarwal, Asghar Ghorbanpour, Masoud Khalkhali, Jiyuan Lu, Balázs Németh, C. Shijia Yu

Abstract: In this paper, we prove a discrete analog of the Selberg Trace Formula for the group GL 3 ( F q ) . By considering a cubic extension of the finite field F q , we define an analog of the upper half-space and an action of GL 3 ( 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 ρ = Ind Γ G 1 for G = GL 3 ( F q ) and Γ = GL 3 ( F p ) .