On abelian $\ell$-towers of multigraphs II

Abstract

Let ℓ be a rational prime. Previously, abelian ℓ-towers of multigraphs were introduced which are analogous to Z ℓ -extensions of number fields. It was shown that for a certain class of towers of bouquets, the growth of the ℓ-part of the number of spanning trees behaves in a predictable manner (analogous to a well-known theorem of Iwasawa for Z ℓ -extensions of number fields). In this paper, we give a generalization to a broader class of regular abelian ℓ-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 Z ℓ and then study the special value at s = 1 of the Artin-Ihara L-function ℓ-adically.

On abelian $\ell$-towers of multigraphs III

“…Meanwhile, in [5], the same behavior was shown to be true with a different method in the situation where the base multigraph is an arbitrary simple graph, meaning that it does not contain loops and parallel edges. In this paper, we extend this result to abelian ℓ-towers over arbitrary base multigraphs (not necessarily simple and not necessarily regular) by adapting the strategy of [10] and [17] to the current more general situation.…”

“…In [10], for each a ∈ Z ℓ , a power series P a (T ) ∈ Z ℓ T obtained as an ℓ-adic limit of some shifted Chebyshev polynomials was constructed in order to prove the main result. (Theorem 4.1 in [10].)…”

“…In [10], for each a ∈ Z ℓ , a power series P a (T ) ∈ Z ℓ T obtained as an ℓ-adic limit of some shifted Chebyshev polynomials was constructed in order to prove the main result. (Theorem 4.1 in [10].) Using Proposition 2.1, the main idea of §2 and §3 of [10] can be adapted to construct other power series which we find more convenient to work with in the present paper.…”

“…To every tuple in Z t ℓ (with t ∈ N and not all entries divisible by ℓ) one can associate an abelian ℓ-tower of a bouquet with t loops. It was proved in [10] that the ℓ-adic valuation of the number of spanning trees behaves similarly to the ℓ-adic valuation of the class numbers in Z ℓ -extensions of number fields. More precisely, if κ n denotes the number of spanning trees of the multigraph at the n-th level, then there exist non-negative integers µ, λ, n 0 and an integer ν such that ord ℓ (κ n ) = µℓ n + λn + ν, for n ≥ n 0 .…”

“…the reader about voltage assignments which give a convenient way of constructing abelian ℓ-towers of multigraphs. One can view the derived multigraphs obtained from voltage assignments as generalizations of the Cayley-Serre multigraphs used in [10] and [17]. In §5, we study the special value at u = 1 of the Artin-Ihara L-functions arising in abelian ℓ-towers of multigraphs.…”