Kida's formula for graphs with ramifications

TL;DR

This paper extends Iwasawa theory for graphs by establishing a Kida's formula for ramified coverings, advancing understanding of invariants and growth of spanning trees in graph coverings.

Contribution

It generalizes Kida's formula to ramified coverings in the context of Iwasawa theory for graphs, which was previously known only for unramified cases.

Findings

Kida's formula is proven for ramified graph coverings.

The asymptotic growth of spanning trees is characterized.

Behavior of lambda- and mu-invariants is analyzed in ramified cases.

Abstract

Recently Iwasawa theory for graphs is developing. A significant achievement includes an analogue of Iwasawa class number formula, which describes the asymptotic growth of the numbers of spanning trees for $\mathbb{Z}_p$-coverings of graphs. Moreover, an analogue of Kida's formula concerning the behavior of the $\lambda$- and $\mu$-invariants is obtained for unramified coverings. In this paper, we establish Kida's formula for possibly ramified coverings.