On abelian $\ell$-towers of multigraphs II

TL;DR

This paper generalizes the study of abelian ll-towers of multigraphs, analyzing their spanning tree growth and ll-adic properties of Artin-Ihara L-functions, extending previous results to broader classes.

Contribution

It introduces a broader class of regular abelian ll-towers of bouquets and studies their properties using ll-adic analysis of Chebyshev polynomials and Artin-Ihara L-functions.

Findings

Growth of ll-part of spanning trees is predictable in the generalized setting.

Shifted Chebyshev polynomials form a family of power series with ll-adic coefficients.

Special value at s=1 of Artin-Ihara L-function is studied ll-adically.

Abstract

Let $\ell$ be a rational prime. Previously, abelian $\ell$-towers of multigraphs were introduced which are analogous to $\mathbb{Z}_{\ell}$-extensions of number fields. It was shown that for a certain class of towers of bouquets, the growth of the $\ell$-part of the number of spanning trees behaves in a predictable manner (analogous to a well-known theorem of Iwasawa for $\mathbb{Z}_{\ell}$-extensions of number fields). In this paper, we give a generalization to a broader class of regular abelian $\ell$-towers of bouquets than was originally considered. To carry this out, we observe that certain shifted Chebyshev polynomials are members of a continuously parametrized family of power series with coefficients in $\mathbb{Z}_{\ell}$ and then study the special value at $s=1$ of the Artin-Ihara $L$-function $\ell$-adically.