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On Zeta Functions of Arithmetically Defined Graphs

✍ Scribed by Ortwin Scheja

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 262 KB
Volume: 5
Category: Article
ISSN: 1071-5797
DOI: 10.1006/ffta.1999.0249

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✦ Synopsis

We study the graph X(n) that is de"ned as the "nite part of the quotient (n)!T, with T the Bruhat}Tits tree over % O ((1/¹ )) and (n) the principal congruence subgroup of "G¸(%

We give concrete realizations of the ¸-functions of the "nite part of the hal#ine !T for "nite unitary representations of that factor over (n), n prime. This allows us to give explicit formulae for the zeta function of X(n) for small n. As an application, we show that these graphs are very good concentrators. Moreover, we construct a new unbounded family of Ramanujan graphs, considering regularizations of X(n).

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