Selberg zeta functions over function fields (Q5950936)

The aim of the paper under review is to elaborate the Selberg trace formula and the Selberg zeta-function for principal congruence subgroups \(\Gamma\) of the group \(\Gamma(1):= \text{PGL} (2,\mathbb{F}_q(t))\). In view of the analogy between number fields and function fields over \(\mathbb{F}_q\) the group \(\Gamma(1)\) may be regarded as an analog of the rational modular group \(\text{PSL} (2,\mathbb{Z})\). \(\Gamma(1)\) acts on the so-called Bruhat-Tits tree \(X\), and \(\Gamma\) defines an infinite arithmetic graph. The adjacency operator on \(X\) induces an operator \(T_\Gamma\) on \(\Gamma\setminus X\) which serves as an analog of the Laplacian, and the usual concepts of Selberg's theory such as point-pair invariants, Selberg transform and Eisenstein series have their natural analogs. The operator \(T_\Gamma\) has both a discrete and a continuous spectrum, and the continuous spectrum is described in terms of Eisenstein series. The various terms in the Selberg trace formula are explicitly computed, and the final results are collected in Theorem 3.2. NEWLINENEWLINENEWLINEApplying the trace formula to a suitable test function, the author introduces the analog of the Selberg zeta-function and determines the relation between the zeta-function and the determinant function for \(T_\Gamma\). The Selberg zeta-function for the principal congruence subgroup \(\Gamma(t)\) is explicitly computed.