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A Survey of Discrete Trace Formulas

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Emerging Applications of Number Theory

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 109))

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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.

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Authors and Affiliations

  1. Mathematics Department, University of California at San Diego, La Jolla, CA, 92093-0112, USA

    Audrey Terras

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  1. Audrey Terras
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Editors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Dennis A. Hejhal

  2. Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada

    Joel Friedman

  3. IBM Thomas J. Watson Research Center, Yorktown Heights, NY, 10598, USA

    Martin C. Gutzwiller

  4. AT&T Labs—Research, Florham Park, NJ, 07932-0971, USA

    Andrew M. Odlyzko

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Terras, A. (1999). A Survey of Discrete Trace Formulas. In: Hejhal, D.A., Friedman, J., Gutzwiller, M.C., Odlyzko, A.M. (eds) Emerging Applications of Number Theory. The IMA Volumes in Mathematics and its Applications, vol 109. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1544-8_28

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  • DOI: https://doi.org/10.1007/978-1-4612-1544-8_28

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7186-4

  • Online ISBN: 978-1-4612-1544-8

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