Tree-lattice zeta functions and class numbers

TL;DR

This paper extends Ihara zeta functions to non-compact arithmetic quotients of Bruhat-Tits trees, revealing their rationality and deriving a prime geodesic theorem that informs class number asymptotics.

Contribution

It introduces a new zeta function for non-compact quotients, establishes its properties, and connects it to class number asymptotics in global fields.

Findings

The zeta function is rational despite infinite-dimensional setting.

Determinant formulas hold with limits of finite minors.

A prime geodesic theorem is derived for these structures.

Abstract

The theory of Ihara zeta functions is extended to non-compact arithmetic quotients of Bruhat-Tits trees. This new zeta function turns out to be a rational function, despite the infinite-dimensional setting. In general it has zeros and poles, in contrast to the compact case. The determinant formulas of Bass and Ihara hold true if one defines the determinant as limit of all finite principal minors. From this analysis, a prime geodesic theorem is derived, which, applied to special arithmetic groups, yields new asymptotic assertions on class numbers of orders in global fields.