Spectra of arithmetic infinite graphs and their application (Q2780953)

This paper surveys the author's earlier papers [Selberg zeta functions over function fields, J. Number Theory 90, 207-238 (2001; Zbl 0992.11053) and The distribution of eigenvalues of arithmetic infinite graphs, Forum Math. 14, 807-829 (2002; Zbl 1142.11334)] and then presents the details of a proof of the prime geodesic theorem for the \((q+1)\)-regular tree \(T\) modulo a congruence subgroup \(\Gamma\)(A) of PGL\((2,F[t])\), where \(F\) denotes a finite field with \(q\) elements. This proof uses standard methods of number theory since the Ihara-Selberg zeta function here is rational and satisfies the Riemann hypothesis. The quotient \(\Gamma/T\) is a finite volume infinite graph known as a Ramanujan diagram [see \textit{M. Morgenstern}, SIAM J. Discrete Math. 7, 560-570 (1994; Zbl 0811.05045)]. NEWLINENEWLINENEWLINEThe paper also considers the interpretation of primitive hyperbolic conjugacy classes in \(\Gamma(1\)) in terms of fundamental units of an order in a quadratic extension of \(F(t)\), which is the function field analogue of a real quadratic extension of the rational numbers.