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arXiv · 2606.03579

An Iwasawa-type asymptotic formula for multiple $\mathbb{Z}_p$-coverings of graphs

Takenori Kataoka
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Abstract

For a possibly ramified $\mathbb{Z}_p^d$-covering of connected graphs, we establish an Iwasawa-type asymptotic formula for the growth of the $p$-adic valuations of the complexities. The formula is expressed as a polynomial in $n$ and $p^n$ with explicit leading coefficients $\lambda$ and $\mu$; in particular, we eliminate the error term of the form $O(p^{(d-1)n})$ appearing in earlier work. We then establish a Kida-type formula describing the behavior of $\lambda$ under a $p$-covering between $\mathbb{Z}_p^d$-coverings, assuming $\mu = 0$. Finally, for any fixed $p$ and integer $d \geq 2$, we construct an unramified $\mathbb{Z}_p^d$-covering of a bouquet with prescribed $\lambda$- and $\mu$-invariants.

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Takenori Kataoka. 2026-06-02. An Iwasawa-type asymptotic formula for multiple $\mathbb{Z}_p$-coverings of graphs. https://arxiv.org/abs/2606.03579

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