Open Access
Feb. 2000 On arithmetic infinite graphs
Hirofumi Nagoshi
Proc. Japan Acad. Ser. A Math. Sci. 76(2): 22-25 (Feb. 2000). DOI: 10.3792/pjaa.76.22


We compute explicitly the Selberg trace formula for principal congruence subgroups $\Gamma$ of $PGL(2, \mathbf{F}_q[t])$, which is the modular group in positive characteristic cases. It is known that $\Gamma \backslash X$ is an infinite Ramanujan diagram, where $X$ is the $q + 1$-regular tres. We express the Selberg zeta function for $\Gamma$ as the determinant of the adjacency operator which is composed of both discrete and continuous spectra. They are rational functions in $q^{-s}$. We also discuss the limit distribution of eigenvalues of $\Gamma \backslash X$ as the level tends to infinity.


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Hirofumi Nagoshi. "On arithmetic infinite graphs." Proc. Japan Acad. Ser. A Math. Sci. 76 (2) 22 - 25, Feb. 2000.


Published: Feb. 2000
First available in Project Euclid: 23 May 2006

zbMATH: 0990.11057
MathSciNet: MR1752819
Digital Object Identifier: 10.3792/pjaa.76.22

Primary: 11F72 , 11M36
Secondary: 05C50 , 58J50

Keywords: function field , graph spectra , Ihara-Selberg zeta function , Ramanujan graph/diagram , Selberg trace formula

Rights: Copyright © 2000 The Japan Academy

Vol.76 • No. 2 • Feb. 2000
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