Iwasawa theory for abelian towers of digraphs

TL;DR

This paper develops an Iwasawa-theoretic framework for abelian $\mathbb{Z}_p^d$-towers of digraphs, linking the $\ell$-primary parts of Picard and Bowen–Franks groups to natural $p$-adic invariants $L_p(X,\alpha)$ and $\mathcal{L}_p(X,\alpha)$. By constructing these $p$-adic zeta and Bowen–Franks functions from voltage assignments and studying them via multi-variable Iwasawa algebras $\Lambda_l(\Gamma)$, the authors prove main conjectures that connect algebraic growth to analytic data, and prove Sinnott–Washington-type growth formulas in the multivariable setting. They show that the Picard groups form torsion Sinnott modules and provide precise asymptotics for the growth of torsion in $\text{Pic}_l(X_n)$ and in Bowen–Franks groups when suitable nonvanishing conditions hold. The defect $\delta(X)=a(X)-b(X)$ is introduced to compare analytic and algebraic ranks, shown to be monotone along towers and often stabilizing, with robust invariance results in isogeny-graph contexts. Overall, the work extends Iwasawa theory to a combinatorial/dynamical setting, connecting graph zeta theory with algebraic invariants and paving the way for applications to isogeny graphs and related dynamics.

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.

Theorems & Definitions (90)

• Theorem A: Theorem \ref{['thm:IMC']}
• Theorem B: Theorem \ref{['thm:generalizedSinnott']}
• Theorem C: Theorem \ref{['thm.growth-delta']}
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Remark 2.6
• Definition 2.7