On $\mathbb{Z}_p$-towers of graph coverings arising from a constant voltage assignment
Antonio Lei, Katharina Müller
TL;DR
This work develops a comprehensive framework for $\mathbb{Z}_p$-towers of graph coverings arising from constant $\mathbb{Z}_p$-valued voltage assignments on finite directed graphs, proving existence and uniqueness of canonical towers and expressing their Iwasawa invariants $\mu(X)$ and $\lambda(X)$ via the determinant $M_X(T)$. It provides precise criteria linking cycle-weight conditions to tower existence, and derives sharp vanishing/non-vanishing results for $\mu(X)$ under structural hypotheses (balanced, regular, volcanic graphs), with concrete applications to (supersingular and ordinary) isogeny graphs and volcanoes. The paper also establishes a non-commutative Iwasawa-theoretic perspective through the $\mathfrak{M}_H(G)$-property in the supersingular setting and demonstrates how these combinatorial invariants reflect arithmetic-geometric features of isogeny graphs. Collectively, the results yield explicit invariant calculations (often $\mu(X)=0$, $\lambda(X)=1$) for broad graph families that model elliptic-curve isogeny graphs, offering a bridge between graph-theoretic voltage-tower constructions and Iwasawa theory. The methods provide a robust toolkit for analyzing how voltage-driven graph coverings encode arithmetic invariants in both standard and non-commutative Iwasawa contexts.
Abstract
We investigate properties of $\mathbb{Z}_p$-towers of graph coverings that arise from a constant voltage assignment. We prove the existence and uniqueness (up to isomorphisms) of such towers. Furthermore, we study the Iwasawa invariants of these towers, and apply our results to towers of isogeny graphs enhanced with level structures, as well as towers arising from volcano graphs.