On abelian $\ell$-towers of multigraphs II

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.