Iwasawa theory for branched double struck upper Z Subscript p Z p Zₚ-towers of finite graphs and Ihara zeta and L -functions

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 whether 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 double struck upper Z Subscript p Z p Zₚ -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.