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.
@article{1148393583,
author = {Nagoshi, Hirofumi},
title = {On arithmetic infinite graphs},
journal = {Proc. Japan Acad. Ser. A Math. Sci.},
volume = {76},
number = {10},
year = {2000},
pages = { 22-25},
language = {en},
url = {http://dml.mathdoc.fr/item/1148393583}
}
Nagoshi, Hirofumi. On arithmetic infinite graphs. Proc. Japan Acad. Ser. A Math. Sci., Tome 76 (2000) no. 10, pp. 22-25. http://gdmltest.u-ga.fr/item/1148393583/