A Selberg Trace Formula for GL₃(<inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">F</mi><mi mathvariant="bold-italic">p</mi></msub></mrow></semantics></math></inline-formula>)∖GL₃(<inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">F</mi><mi mathvariant="bold-italic">q</mi></msub></mrow></semantics></math></inline-formula>)/<i>K</i>

Abstract

In this paper, we prove a discrete analog of the Selberg Trace Formula for the group GL3(Fq). By considering a cubic extension of the finite field Fq, we define an analog of the upper half-space and an action of GL3(Fq) 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=IndΓG1 for G=GL3(Fq) and Γ=GL3(Fp).

Keywords

• general linear group over finite fields
• finite upper half space
• pre-trace formula
• Selberg type trace formula
• orbital sums