Iwasawa theory for branched ${\mathbb{Z}}_p$-towers of finite graphs

Abstract

We initiate the study of Iwasawa theory for branched ${\mathbb{Z}}_p$-towers of finite connected graphs. These towers are more general than what have been studied so far, since the morphisms of graphs involved are branched covers, a particular kind of harmonic morphisms of graphs. We prove an analogue of Iwasawa’s asymptotic class number formula for the $p$-part of the number of spanning trees in this setting. Moreover, we find an explicit generator for the characteristic ideal of the finitely generated torsion Iwasawa module governing the growth of the $p$-part of the number of spanning trees in such towers.