2021/07/15 by McGown, Kevin J., Vallières, Daniel
#05C25 #11R18 #11Z05 #13F20 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2107.07639
Let ℓ be a rational prime. Previously, abelian ℓ-towers of multigraphs were introduced which are analogous to \Zℓ-extensions of number fields. It was shown that for 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 \Zℓ-extensions of number fields). In this paper, we extend this result to abelian ℓ-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 \Z_ℓ 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.