On abelian $$\ell $$-towers of multigraphs II

Abstract

Let be a rational prime. Previously, abelian -towers of multigraphs were introduced which are analogous to Z -extensions of number fields. It was shown that for a certain class of towers of bouquets, the growth of the -part of the number of spanning trees behaves in a predictable manner (analogous to a well-known theorem of Iwasawa for Z -extensions of number fields). In this paper, we give a generalization to a broader class of regular abelian -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 and then study the special value at u = 1 of the Artin-Ihara L-function -adically. RésuméSoit un nombre premier rationnel. Dans un article précédent, les -tours abéliennes de multigraphes furent introduites qui peuvent être vues comme étant analogues aux Z -extensions de corps de nombres. Il fut démontré que pour une certaine famille de tours de bouquets, la valuation -adique du nombre d'arbres couvrants varie de façon prévisible (de manière analogue à un théorème connu d'Iwasawa pour les Z -extensions de corps de nombres). Dans cet article, nous généralisons ce résultat à une famille plus large de -tours régulières et abéliennes de bouquets que celles considérées précédemment. Afin d'accomplir ce but, nous observons que certains polynômes de Tchebychev décalés sont membres d'une famille continûment paramétrée de séries entières à coefficients dans Z et étudions ensuite la valeur spéciale en u = 1 des fonctions L d'Artin-Ihara -adiquement.

On the BDP Iwasawa main conjecture for modular forms

“…it can again be described completely in terms of some invariants µ, λ and ν attached to the Z p -cover X ∞ /X. The proof of this result, which was given in increasing generality in a series of papers of Vallières and McGown (see [37,25,26]), was completely analytic, in the sense that it was based on a direct relation of the number of spanning trees to certain special values of Artin-Ihara L-functions, which can be computed explicitly in this setting. In [8] the above approach was generalised to Z l p -covers of graphs.…”

On the Distribution of Iwasawa Invariants Associated to Multigraphs

“…It is shown by McGown and the fourth named author (cf. [19], [20], [26]) that there are invariants μ, λ ∈ Z ≥0 and ν ∈ Z such that κ (X n ) = (μ n +λ +ν)…”

“…The following result shows that f X,α (T ) = 0 under the assumptions imposed on α. The proof is essentially contained in [19], [20], and [26], we recall these details. First, we introduce some further notation.…”

“…Remark 4.7. Our numerical calculations suggest that for a given , the right-hand side of the inequality given by Theorem 4.6 tends to In what follows, we prove a necessary and sufficient condition for μ > 0 in the special case where α k ∈ Z for all k. Given an integer a ∈ Z, let us write d k (a) for the coefficient of X k in the shifted Chebyshev polynomial P a (X) used in [19,Sec. 2].…”

“…What might be called the Iwasawa theory of multigraphs is a recent area of research. The reader may refer to [19], [20], and [26] where the approach via Artin–Ihara L -functions is introduced and to [8] where the module theoretical approach was initiated. Other articles on the subject where the reader can find more information are [3], [12], [14], [16], and [22].…”