On abelian \(\ell \)-towers of multigraphs III

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 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 extend this result to abelian \(\ell \)-towers over an arbitrary connected multigraph (not necessarily simple and not necessarily regular). In order to carry this out, we employ integer-valued polynomials to construct power series with coefficients in \(\mathbb {Z}_\ell \) arising from cyclotomic number fields, different than the power series appearing in the prequel. This allows us to study the special value at \(u=1\) of the Artin–Ihara L-function, when the base multigraph is not necessarily a bouquet.