Ihara zeta functions and class numbers

TL;DR

This paper connects Ihara zeta functions with class number asymptotics of orders in imaginary quadratic fields using prime geodesic theorems in Bruhat-Tits buildings.

Contribution

It introduces a novel application of prime geodesic theorems to derive new asymptotic results on class numbers in algebraic number theory.

Findings

Derived new asymptotic formulas for class numbers

Applied prime geodesic theorem to division algebra unit groups

Established connections between geometric and number theoretic properties

Abstract

The prime geodesic theorem for cycles in Bruhat-Tits buildings is applied to unit groups of division algebras to derive new asymptotic assertion on class numbers of orders in imaginary quadratic fields.