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 \(u=1\) of the Artin-Ihara L-function \(\ell \)-adically.