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arXiv · 2201.05186

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

Antonio Lei·Daniel Vallières
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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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Antonio Lei, Daniel Vallières. 2023-01-24. The non-$\ell$-part of the number of spanning trees in abelian $\ell$-towers of multigraphs. https://arxiv.org/abs/2201.05186

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