Axioms, Volume 13, Issue 6 (June 2024) – 78 articles
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The Selberg trace formula is a celebrated theorem relating spectral theory, geometry and number theory. In its original form, it relates the spectrum of the Laplacian on a hyperbolic surface to lengths of closed geodesics on the surface. It can be viewed as a non-Abelian generalization of the Poisson summation formula in Fourier analysis. In this paper, we prove an analogue of this result for general linear groups of 3 by 3 matrices over finite fields by defining an analogue of the Poincare upper half plane for general linear groups over finite fields. We give explicit formulas for orbital sums defined by hyperbolic, elliptic and parabolic elements in the group. View this paper
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