zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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].
11F72Spectral theory; Selberg trace formula
11T60Finite upper half-planes
05C25Graphs and abstract algebra