Abstract
The goal is to survey work on Selberg’s trace formula for discrete quotient spaces G/K both finite and infinite. Here G is often the general linear group GL(n, F) consisting of n × n non-singular matrices with entries in some field F, and K is some subgroup. Usually F is the finite field F q with q elements and n = 2.We begin with the trace formula for finite abelian groups (i.e., Poisson’s summation formula) and an application to error-correcting codes. For non-abelian groups, we consider three main topics: • an application of the pre-trace formula to find some isospectral non- isomorphic Schreier graphs with vertex sets GL(3, F2)/Г i , i = 1, 2, with Г1 consisting of matrices having first column equal to \(\left( {\begin{array}{*{20}{c}} 1 \\ 0 \\ 0 \\ \end{array} } \right)\) and Г2 the transpose of Г1; • the trace formula for GL(2, F q )/GL(2, F p ), where q = p r; • the trace formula on the k-regular tree (which is a p-adic quotient space if k = p + 1) and Ihara’s theorem for the zeta function of a finite k- regular graph. KeywordsSymmetric SpaceConjugacy ClassZeta FunctionHeat KernelHalf PlaneThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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