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

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 and such that for . This establishes an analog of Iwasawa’s formula (1.1) in the setting of multigraphs.…”

“…18.15] for the Artin–Ihara L -function gives where is the twisted adjacency matrix of Definition 2.9, D is the valency matrix of X , and is the Euler characteristic of X . From now on, we let Coming back to the abelian -tower associated with , for all positive integer n , [20, Cor. 5.6] implies where is the character of satisfying .…”

“…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].…”

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 ℤ ℓ d -towers of graphs

“…More recently, an analogous result was proven in the context of graph theory where field extensions are replaced with graph covers and the class number is replaced with the number of spanning trees. See [3,4,12,13,29].…”