Understanding ramification of branched {$\mathbb{Z}_p$}-covers

TL;DR

This paper introduces a combinatorial method to analyze how ramification in branched {$ obreak ext{Z}_p$}-covers influences the enumeration of spanning trees, connecting graph structure to Iwasawa invariants.

Contribution

It presents a novel combinatorial framework involving segments and segmental decompositions to study ramification effects on spanning tree counts in branched {$ ext{Z}_p$}-covers.

Findings

Segmental decomposition explains ramification impact on spanning trees.

Method links graph structure to Iwasawa invariants.

Provides a new combinatorial perspective on branched covers.

Abstract

We provide a combinatorial approach to counting the number of spanning trees at the $n$-th layer of a branched $\mathbb{Z}_p$-cover of a finite connected graph $\mathsf{X}$. Our method achieves in explaining how the position of the ramified vertices affects the count and hence the Iwasawa invariants. We do so by introducing the notion of segments, segmental decomposition of a graph, and number of segmental $t$-tree spanning forests.