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A survey of discrete trace formulas. (English) Zbl 0982.11031
Hejhal, Dennis A. (ed.) et al., Emerging applications of number theory. Based on the proceedings of the IMA summer program, Minneapolis, MN, USA, July 15-26, 1996. New York, NY: Springer. IMA Vol. Math. Appl. 109, 643-681 (1999).
The aim of the work under review is to survey work on discrete analogues of the Selberg trace formula. In the introduction, the author studies the trace formula (Poisson summation formula) on finite abelian groups and its application to the MacWilliams identity relating the weight enumerator polynomial of a code and its dual. The second section deals with the pre-trace formula and isospectral Schreier graphs for some non-abelian finite groups. The third section elaborates on the trace formula for $\text{GL}(2,\bbfF_p)$ acting on a finite upper half-plane defined over the field $\bbfF_q$ where $q=p^r$. In the last section the trace formula for the $(q+1)$-regular tree is used to study the spectra of adjacency operators of $(q+1)$-regular graphs as well as Ihara’s formula for the zeta functions of such graphs. -- This survey paper may serve as a well-motivated guide to the literature and gives numerous useful references. For the entire collection see [Zbl 0919.00047].
##### MSC:
 11F72 Spectral theory; Selberg trace formula 11T60 Finite upper half-planes 05C05 Trees 05C25 Graphs and abstract algebra