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The non-\(\ell \)-part of the number of spanning trees in abelian \(\ell \)-towers of multigraphs

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Abstract

Let \(\ell \) and p be two distinct primes. We study the p-adic valuation of the number of spanning trees in an abelian \(\ell \)-tower of connected multigraphs. This is analogous to the classical theorem of Washington–Sinnott on the growth of the p-part of the class group in a cyclotomic \({\mathbb {Z}}_\ell \)-extension of abelian extensions of \({\mathbb {Q}}\). Furthermore, we show that under certain hypotheses, the number of primes dividing the number of spanning trees is unbounded in such a tower.

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Acknowledgements

The authors would like to thank Cédric Dion and Anwesh Ray for interesting discussions on topics related to the present article. DV would like to thank the pure mathematics group at the California State University - Chico including John Lind, Thomas Mattman, and Kevin McGown for several stimulating discussions during our seminar. DV would also like to thank AL for inviting him to give a talk at the Université Laval on related topics from which the current paper is a result of discussions that followed. AL’s research is supported by the NSERC Discovery Grants Program RGPIN-2020-04259 and RGPAS-2020-00096. AL would like to thank DV for introducing this beautiful subject to him.

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  1. Department of Mathematics and Statistics, University of Ottawa, 150 Louis-Pasteur Pvt, Ottawa, ON, K1N 6N5, Canada

    Antonio Lei

  2. Mathematics and Statistics Department, California State University, Chico, CA, 95929, USA

    Daniel Vallières

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  1. Antonio Lei
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  2. Daniel Vallières
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Correspondence to Antonio Lei.

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Lei, A., Vallières, D. The non-\(\ell \)-part of the number of spanning trees in abelian \(\ell \)-towers of multigraphs. Res. number theory 9, 18 (2023). https://doi.org/10.1007/s40993-023-00425-1

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