Iwasawa theory for abelian towers of digraphs

Abstract

Let $p$ and $\ell$ be prime numbers, and $d\ge1$ an integer. We formulate and prove Iwasawa main conjectures of the Picard groups and Bowen--Franks groups in $\mathbb{Z}_p^d$-towers of digraphs. In particular, we relate the $\ell$ parts of these groups to certain $p$-adic $L$-functions defined using a voltage assignment. In the case where $\ell$ is not equal to $p$, we make use of the recent work of Bandini--Longhi to define the appropriate characteristic ideals. We also prove the growth of the $\ell$-part of these groups, generalizing classical results of Sinnott and Washington on ideal class groups of number fields. Finally, we introduce the concept of defect, which compare certain algebraic and analytic ranks related to Bowen--Franks groups and study their asymptotic behaviour in a $\mathbb{Z}_p^d$-tower.