# $p$-adic Mahler measure and $\mathbb{Z}$-covers of links

## TL;DR

This paper develops a $p$-adic Mahler measure theory and applies it to analyze the growth of torsion homology in $Z$-covers of links, revealing new relationships among Alexander polynomials, $p$-adic entropy, and Iwasawa invariants.

## Contribution

It introduces a $p$-adic Mahler measure framework and establishes a $p$-adic analogue of torsion homology growth formulas for link covers.

## Key findings

- Established a $p$-adic asymptotic formula for torsion homology growth.
- Derived a balance formula linking Alexander polynomial, $p$-adic entropy, and Iwasawa $ ext{μ}_p$-invariant.
- Applied $p$-adic theory to various examples of link covers.

## Abstract

Let $p$ be a prime number. We develop a theory of $p$-adic Mahler measure of polynomials and apply it to the study of $\mathbb{Z}$-covers of rational homology 3-spheres branched over links. We obtain a $p$-adic analogue of the asymptotic formula of the torsion homology growth and a balance formula among the leading coefficient of the Alexander polynomial, the $p$-adic entropy, and the Iwasawa $\mu_p$-invariant. We also apply the purely $p$-adic theory of Besser--Deninger to $\mathbb{Z}$-covers of links. In addition, we study the entropies of profinite cyclic covers of links. We examine various examples throughout the paper.