Iwasawa theory for branched $\mathbb{Z}_{p}$-towers of finite graphs and Ihara zeta and $L$-functions

TL;DR

This paper explores Iwasawa theory for branched $ ext{Z}_p$-towers of finite graphs, establishing analogues of classical number theory results using Ihara zeta and $L$-functions, and analyzing their properties.

Contribution

It extends Iwasawa theory to graph towers, proving an analogue of class number growth and relating Ihara $L$-functions to characteristic ideals of Iwasawa modules.

Findings

Proves the analogue of Iwasawa's class number formula for graph towers.

Establishes the Artin formalism for Ihara $L$-functions in this context.

Connects the characteristic ideal generator with Ihara $L$-functions.

Abstract

We revisit the theory of Ihara $L$-functions in the context initially studied by Bass and Hashimoto and more recently by Zakharov. In particular, we study if the Artin formalism is satisfied by these $L$-functions. As an application, we give a proof of the analogue of Iwasawa's asymptotic class number formula for the $p$-part of the number of spanning trees in branched $\mathbb{Z}_{p}$-towers of finite connected graphs using Ihara zeta and $L$-functions. Moreover, we relate a generator for the characteristic ideal of the finitely generated torsion Iwasawa module that governs the growth of the $p$-part of the number of spanning trees in such towers with Ihara $L$-functions.