$p$-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.

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.

Equations

$$\log \text{textsc}{m}_p(f(z)):=\underset{n\in \mathbb{N};\gcd (n,p)=1}{\lim }\frac{1}{n}\sum _{{\zeta }^n=1}\log |f(\zeta ){|}_p,$$

$$-\log |a_0{|}_p=h_p+{\mu }_p\log p.$$

$$\log \text{textsc}m(f(t))=\frac{1}{2\pi \sqrt{-1}}{\int }_{|z|=1}\log |f(z)|\frac{dz}{z}.$$

$$\log \text{textsc}m(f(t))$$

$$=\log |a_0|+\sum _{1<|{\alpha }_i|}\log |{\alpha }_i|.$$

$$R(f(t),g(t)):=a^{\mathrm{deg}(g)}{b}^{\mathrm{deg}(f)}\prod _{i,j}({\alpha }_i-{\beta }_j).$$

$$R(f(t),g(t))$$

$$R(f(t),g(t)h(t))$$

$$\underset{n\in \mathbb{N};\text{value}\ne 0}{\lim }|R(f(t),{\nu }_n(t)){|}^{1/n}=\text{textsc}m(f(t)).$$

$${\int }_{|z{|}_p=1}F(z)\frac{dz}{z}:=\underset{n\in \mathbb{N};\gcd (n,p)=1}{\lim }\frac{1}{n}\sum _{{\zeta }^n=1}F(\zeta ),$$

$$\log \text{textsc}{m}_p(f(t))=\underset{n\in \mathbb{N};\gcd (n,p)=1}{\lim }\frac{1}{n}\sum _{{\zeta }^n=1}\log |f(\zeta ){|}_p,$$

$$\log \text{textsc}{m}_p(f(t))$$

$$=\log |a_0{|}_p+\sum _{1<|{\alpha }_i{|}_p}\log |{\alpha }_i{|}_p.$$

$$\log \text{textsc}{m}_p(f(t))=\underset{n\in \mathbb{N}}{\lim }\frac{1}{n}\sum _{{\zeta }^n=1;f(\zeta )\ne 0}\log |f(\zeta ){|}_p,$$

$$\underset{n\in \mathbb{N};\text{value}\ne 0}{\lim }|R(f(t),{\nu }_n(t)){|}_p^{1/n}=\text{textsc}{m}_p(f(t)).$$

$$\prod _{{\zeta }^n=1;f(\zeta )\ne 0}|\zeta -\alpha {|}_p=\frac{\prod _{{\zeta }^n=1}|\zeta -\alpha {|}_p}{\prod _{{\zeta }^n=1;f(\zeta )=0}|\zeta -\alpha {|}_p}.$$

$$\prod _{{\zeta }^n=1;f(\zeta )\ne 0}|\zeta -\alpha {|}_p=\frac{\prod _{{\zeta }^n=1;\zeta \ne \alpha }|\zeta -\alpha {|}_p}{\prod _{{\zeta }^n=1;f(\zeta )=0,\zeta \ne \alpha }|\zeta -\alpha {|}_p}.$$

$$\underset{n\in \mathbb{N}}{\lim }\prod _{{\zeta }^n=1;f(\zeta )\ne 0}|f(\zeta ){|}_p^{1/n}=|a_0{|}_p\prod _i\max \{|{\alpha }_i{|}_p,1\}$$

$$A_{L}(t)=(t-1)^{d-1}H_{L}(t)=\left\{\begin{array}[]{ll}\Delta_{L}(t)&{\rm if}\ d=1,\\ (t-1)\Delta_{L}(t)&{\rm if}\ d>1.\end{array}\right.$$

$$|H_1(M_n)|=|R(A_L(t),{\nu }_n(t))|\times C$$

$$\underset{n\in \mathbb{N};\text{value}\ne 0}{\lim }|H_1(M_n){|}^{1/n}=\text{textsc}m(A_L(t)),$$

$$\underset{n\in \mathbb{N};\text{value}\ne 0}{\lim }||H_1(M_n)|{|}_p^{1/n}=\text{textsc}{m}_p(A_L(t)).$$

$$||H_1(M_{p^r})|{|}_p^{-1}=p^{{\lambda }_{p}r+{\mu }_p{p}^r+{\nu }_p}$$

$$\log \text{textsc}{m}_p(A_L(t))=-{\mu }_p\log p.$$

$$h=\sum _{p\le \infty }{h}_p,h_p=\log \text{textsc}{m}_p(B(t)),$$

$$h=\sum _{p\le \infty }{h}_p,h_p=\log \text{textsc}{m}_p(A_L(t)/a_0).$$

$$-\log |a_0{|}_p=h_p+{\mu }_p\log p.$$

$$\log |a_0|=\sum _{p<\infty }\sum _{|{\alpha }_i{|}_p>1}\log |{\alpha }_i{|}_p+\sum _{p<\infty }{\mu }_p\log p.$$

$$m_p(f(z)):={\int }_{|z{|}_p=1}{\log }_{p}f(z)\frac{dz}{z}:=\underset{n\in \mathbb{N};\gcd (n,p)=1}{\lim }\sum _{{\zeta }^n=1}{\log }_{p}f(z),$$

$$m_p(f(t))$$

$$={\log }_p{a}_0+\sum _{|{\alpha }_i{|}_p>1}{\log }_p{\alpha }_i.$$

$$\underset{n\in \mathbb{N};(n,p)=1}{\lim }\frac{1}{n}{\log }_p|R(f(t),t^n-1)|=m_p(f(t)).$$

Full text

1. 1 Introduction

2. 2 $p$-adic Mahler measure of polynomials

3. 2.1 Euclidean Mahler measure

4. 2.2 $p$-adic Mahler measure $\textsc{m}_{p}$

5. 2.3 Proofs

6. 2.4 Alternative modification of the integral path

7. 3 $p$-adic Mahler measure and ${\mathbb{Z}}$-covers of links

8. 3.1 ${\mathbb{Z}}$-covers and $\textsc{m}_{p}$

9. 3.2 Iwasawa $\mu_{p}$-invariant and $\textsc{m}_{p}$

10. 3.3 $p$-adic entropy $h_{p}$ and $\textsc{m}_{p}$

11. 3.4 $h_{p}$, $\mu_{p}$, and the leading coefficient $a_{0}$

12. 4 Purely $p$-adic notions

13. 4.1 Besser–Deninger $m_{p}$

14. 4.2 Deninger’s $\hbar_{p}$

15. 5 Profinite cyclic covers

16. 5.1 ${\mathbb{Z}}_{p}$-covers

17. 5.2 ${\widehat{{\mathbb{Z}}}}$-covers

1. Introduction

It is well known that the asymptotic behavior of the torsion homology growth in a ${\mathbb{Z}}$-cover of $S^{3}$ branched over a link is described by the Euclidean Mahler measure of the reduced Alexander polynomial of the branch link.

The Euclidean Mahler measure $\textsc{m}(f(t))$ of a Laurent polynomial $0\neq f(t)\in{\mathbb{C}}[t^{\pm 1}]$ is defined by $\displaystyle\log\textsc{m}(f(t))=\frac{1}{2\pi\sqrt{-1}}\int_{|z|=1}\log|f(z)|\frac{dz}{z}$. If $\displaystyle f(t)=a\prod_{i}(t-\alpha_{i})$, then we have Jensen’s formula $\displaystyle\textsc{m}(f(t))=|a|\prod_{i}{\rm max}\{|\alpha_{i}|,1\}$ ([EW99]).

If $H$ is a finite group, then we denote its order by $|H|$. If $H$ is an infinite group, then we put $|H|=0$. For two polynomials $f(t)$ and $g(t)$ in ${\mathbb{Z}}[t]$, we denote their resultant by $R(f(t),g(t))$. Now let $M$ be a rational homology 3-sphere (${\mathbb{Q}}$HS3) and let $L$ be a link in $M$. Let $h_{\infty}:X_{\infty}\to X:=M-L$ be a ${\mathbb{Z}}$-cover over the exterior with a generator $t$ of the deck transformation group and let $\{h_{n}:M_{n}\to M\}_{n}$ denote the system of branched ${\mathbb{Z}}/n{\mathbb{Z}}$-covers over $(M,L)$ obtained from subcovers of $h_{\infty}$ by the Fox completion. The Alexander module $H_{1}(X_{\infty})$ is a finitely generated ${\mathbb{Z}}[t^{\mathbb{Z}}]$-module. We take a generator $A_{L}(t)$ of the maximum principal ideal containing the Fitting ideal ${\rm Fitt}_{{\mathbb{Z}}[t^{\mathbb{Z}}]}H_{1}(X_{\infty})$ of $H_{1}(X_{\infty})$ over ${\mathbb{Z}}[t^{{\mathbb{Z}}}]$ with $A_{L}(t)\in{\mathbb{Z}}[t]\subset{\mathbb{Z}}[t^{\mathbb{Z}}]$ and call it an/the Alexander polynomial of $h_{\infty}$. Put $\nu_{n}(t):=(t^{n}-1)/(t-1)$. Then by a generalization of Fox’s formula (Proposition 3.2, e.g. [KM08]), there exists some finite set $\mathcal{C}\subset{\mathbb{N}}$ such that for any $n\gg 0$ the resultant $R(A_{L}(t),\nu_{n}(t))$ satisfies $|H_{1}(M_{n})|=|R(A_{L}(t),\nu_{n}(t))|\times C$ for some $C\in{\mathcal{C}}$. By a similar argument to [GAS91], we obtain an asymptotic formula of the homology growth (Theorem 3.3): $\displaystyle\lim_{n\in{\mathbb{N}};\text{value}\neq 0}|H_{1}(M_{n})|^{1/n}=\textsc{m}(A_{L}(t))$.

For a prime number $p$, let $|\cdot|_{p}$ denote the $p$-adic norm (absolute value) normalized by $|p|_{p}=p^{-1}$. If $H$ is a group, then we write $||H||_{p}=|(|H|)|_{p}$. For an inverse system $\{M_{p^{r}}\to M\}_{r}$ of branched ${\mathbb{Z}}/p^{r}{\mathbb{Z}}$-covers over $(M,L)$, a $p$-adic refinement of the asymptotic formula called the Iwasawa type formula (Proposition 3.6) is known: If $M_{p^{r}}$’s are ${\mathbb{Q}}$HS3’s, then there exist some $\lambda_{p},\mu_{p}\in{\mathbb{N}},\nu_{p}\in{\mathbb{Z}}$ called the Iwasawa invariants such that for any $r\gg 0$ the equation $||H_{1}(M_{p^{r}})||_{p}^{-1}=p^{\lambda_{p}r+\mu_{p}p^{r}+\nu_{p}}$ holds.

The main purpose of this paper is to investigate the relation between the Mahler measure and the Iwasawa invariants.

For a prime number $p$, let ${\mathbb{C}}_{p}={\widehat{\overline{{\mathbb{Q}}}}}_{p}$ denote the $p$-adic completion of an algebraic closure of the $p$-adic number field ${\mathbb{Q}}_{p}$, and fix an immersion $\overline{{\mathbb{Q}}}\hookrightarrow{\mathbb{C}}_{p}$ of an algebraic closure of ${\mathbb{Q}}$. (We regard $p=\infty$ for Euclidean objects.)

We define the $p$-adic Mahler measure $\textsc{m}_{p}(f(z))$ of a Laurent polynomial $f(t)\in{\mathbb{C}}_{p}[t^{\pm 1}]$ ($\neq 0$) by imitating the Shnirel’man integral ([Shn38]) as follows: We first suppose that $f(t)$ does not vanish on the $p$-adic unit circle $|z|_{p}=1$ and put

[TABLE]

where the limit is taken with respect to the Euclidean topology. If $\displaystyle f(t)=\sum_{0\leq i\leq l}a_{i}t^{d-i}=a_{0}t^{d-l}\prod_{0\leq i\leq l}(t-\alpha_{i})$ with $d\in{\mathbb{Z}}$ and $l\in{\mathbb{N}}$, then an analogue $\displaystyle\textsc{m}_{p}(f(t))=|a_{0}|_{p}\prod_{0\leq i\leq l}{\rm max}\{|\alpha_{i}|_{p},1\}$ of Jensen’s formula holds (Theorem 2.3). In addition, the equation $\textsc{m}_{p}(f(t))={\rm max}\{|a_{i}|_{p}\}_{i}$ holds (Proposition 2.7), that is, $\textsc{m}_{p}(f(t))$ coincides with the Gauss norm of $f(t)$.

We extend this notion with Jensen’s formula to the case with roots on $|z|_{p}=1$ and remove the condition ${\rm gcd}(n,p)=1$ on the limit (Theorem 2.5). Then for a ${\mathbb{Z}}$-cover over $(M,L)$ with its polynomial $A_{L}(t)$, we obtain a $p$-adic analogue $\displaystyle\lim_{n\in{\mathbb{N}};\text{value}\neq 0}||H_{1}(M_{n})||_{p}^{1/n}=\textsc{m}_{p}(A_{L}(t))$ of the asymptotic formula (Theorem 3.3). In addition, the formula $\log\textsc{m}_{p}(A_{L}(t))=-\mu_{p}\log p$ of the Iwasawa invariant follows (Proposition 3.7).

Now we consider a ${\mathbb{Z}}$-cover over $(S^{3},L)$ with $\displaystyle A_{L}(t)=\sum_{0\leq i\leq l}a_{i}t^{d-i}=a_{0}t^{d-l}\prod_{0\leq i\leq l}(t-\alpha_{i})$ over $\overline{{\mathbb{Q}}}$. The relation between the Mahler measure and the topological entropy is well known. It is also known that $a_{0}$ has a topological information of $L$. Noguchi studied the relation among them from a viewpoint of $p$-adic topology ([Nog07]). We generalize his work for links.

We consider the meridian action on $H_{1}(X_{\infty},{\mathbb{Q}})$ and its Pontryagin dual. Then by [GBV15] and [LW88], the algebraic/topological entropy is given by $\displaystyle h=\sum_{p\leq\infty}h_{p}=$ $\displaystyle\sum_{p\leq\infty}\log\textsc{m}_{p}(A_{L}(t))$ with use of the $p$-adic entropies  $\displaystyle h_{p}=\log\textsc{m}_{p}(A_{L}(t)/a_{0})=$  $\displaystyle\sum_{|\alpha_{i}|_{p}>1}\log|\alpha_{i}|_{p}$ (Proposition 3.11). We see a balance formula among $a_{0}$, $h_{p}$, and the Iwasawa $\mu_{p}$-invariant (Proposition 3.13):

[TABLE]

By the product formula $\displaystyle\prod_{p\leq\infty}|a_{0}|_{p}=1$, a generalization of [Nog07, Corollary 4] follows: $\displaystyle\log|a_{0}|=\sum_{p<\infty}\sum_{|\alpha_{i}|_{p}>1}\log|\alpha_{i}|_{p}+\sum_{p<\infty}\mu_{p}\log p$ (Corollary 3.15).

Besser and Deninger ([BD99]) defined the purely $p$-adic log Mahler measure $m_{p}$, which is different from ours, with use of the $p$-adic logarithm and the Shnirel’man integral, and proved Jensen’s formula for it. In addition, Deninger ([Den09]) introduced the notion of the purely $p$-adic entropy $\hbar_{p}$ by using the numbers of fixed points and the $p$-adic logarithm, and proved the relation between $m_{p}$ and $\hbar_{p}$. In Section 4, we apply their theory to ${\mathbb{Z}}$-covers of $S^{3}$ branched over links.

Finally, in Section 5, we study profinite cyclic covers of $S^{3}$ branched over links. We consider the $p$-adic integer ring ${\mathbb{Z}}_{p}=\varprojlim_{n}{\mathbb{Z}}/p^{n}{\mathbb{Z}}$ and the profinite integer ring ${\widehat{{\mathbb{Z}}}}=\varprojlim_{n}{\mathbb{Z}}/n{\mathbb{Z}}$. We investigate the entropies of the meridian actions on the Iwasawa modules ${\mathcal{H}}_{p}$ of ${\mathbb{Z}}_{p}$-covers and those on the completed Alexander modules ${\mathcal{H}}$ of ${\widehat{{\mathbb{Z}}}}$-covers for the cases with $A_{L}(t)\in{\mathbb{Z}}[t]$.

    In Sections 3–5, we compute various examples obtained from links in the Rolfsen table ([Rol76]).

2. $p$-adic Mahler measure of polynomials

2.1. Euclidean Mahler measure

We consider the complex function on ${\mathbb{C}}^{*}$ given by a Laurent polynomial $0\neq f(t)\in{\mathbb{C}}[t^{\pm 1}]$. If $\displaystyle f(t)=\sum_{0\leq i\leq l}a_{i}t^{d-i}=a_{0}t^{d-l}\prod_{0\leq i\leq l}(t-\alpha_{i})$ with $a_{0}a_{l}\neq 0$ and it has no root on the unit circle $|z|=1$, then its Euclidean Mahler measure $\textsc{m}(f(t))$ is defined by

[TABLE]

It satisfies Jensen’s formula

[TABLE]

It is generalized to the cases with roots on $|z|=1$. We define $\textsc{m}(f(t))$ as the limit of the integrals along paths avoiding roots on $|z|=1$ [EW99]. Indeed, suppose $f(\alpha)=0,|\alpha|=1$. Then for each simple closed curve $\gamma$ around $\alpha$ with finite length $|\gamma|$, the integration of a bounded continuous function $\log|f(z)|/z$ along $\gamma$ is bounded by $|\gamma|\times(\text{constant})$. Thus we have $\displaystyle\lim_{|\gamma|\to 0}\int_{\gamma}\log|f(z)|\frac{dz}{z}=0$.

For polynomials $f(t)=a\prod_{i}(t-\alpha_{i})$, $g(t)=b\prod_{j}(t-\beta_{j})\in{\mathbb{Z}}[t]$, their resultant is defined as

[TABLE]

It coincides with the determinant of the Sylvester matrix, whose elements are given by their coefficients, and hence satisfies $R(f(t),g(t))\in{\mathbb{Z}}$. We have $R(f(t),g(t))=0$ if and only if they gave no common zero. We have

[TABLE]

for each $h(t)\in{\mathbb{Z}}[t]$. If $f(t)=0$, then we put $R(0,g(t))=0$. If $g(t)\neq 0$, then we put $R(1,g(t))=1$ ([Web79]).

By an argument in [GAS91], we have an asymptotic formula of the resultants:

Proposition 2.1**.**

Let $0\neq f(t)\in{\mathbb{Z}}[t]$ and $\displaystyle\nu_{n}(t)=(t^{n}-1)/(t-1)=\sum_{0\leq i<n}t^{i}$. Then

[TABLE]

In order to prove this, we need to show $\displaystyle\lim_{n\in{\mathbb{N}};\text{value}\neq 0}|\alpha^{n}-1|^{1/n}=1$ for a root $\alpha$ with $|\alpha|=1$. In [GAS91], the notion of ${\rm arg}\alpha$ was used in obtaining the estimation $|\alpha^{n}-1|>C\exp(-(\log n)^{6})$ for a constant $C$.

2.2. $p$-adic Mahler measure $\textsc{m}_{p}$

For a prime number $p$, let $|\cdot|_{p}$ denote the $p$-adic norm normalized by $|p|_{p}=1/p$, and let ${\mathbb{Q}}_{p}$ denote the $p$-adic numbers, that is, the $p$-adic completion of ${\mathbb{Q}}$. Let ${\mathbb{C}}_{p}={\widehat{\overline{{\mathbb{Q}}}}}_{p}$ denote the $p$-adic completion of an algebraic closure of ${\mathbb{Q}}_{p}$, and fix an immersion $\overline{{\mathbb{Q}}}\hookrightarrow{\mathbb{C}}_{p}$ of an algebraic closure of ${\mathbb{Q}}$. For a continuous function $F:\{z\in{\mathbb{C}}_{p}\mid|z|_{p}=1\}\to{\mathbb{C}}_{p}$ with no zero on $|z|_{p}=1$, the Shnirel’man integral is defined by

[TABLE]

where the limit is taken with respect to the $p$-adic topology ([Shn38]).

We imitate this notion to define the $p$-adic Mahler measure as follows:

Definition 2.2**.**

Let $f:{\mathbb{C}}_{p}^{*}\to{\mathbb{C}}_{p}$ be the continuous map defined by a Laurent polynomial $0\neq f(t)\in{\mathbb{C}}_{p}[t^{\pm 1}]$. If $f(t)$ has no root on $|z|_{p}=1$, then we define its $p$-adic Mahler measure $\textsc{m}_{p}(f(t))$ by

[TABLE]

where the limit is taken with respect to the Euclidean topology.

Then we have an analogue of Jensen’s formula:

Theorem 2.3** (Jensen’s formula).**

If $\displaystyle f(t)=\sum_{0\leq i\leq l}a_{i}t^{d-i}=a_{0}t^{d-l}\prod_{0\leq i\leq l}(t-\alpha_{i})\in{\mathbb{C}}_{p}[t^{\pm 1}]$ with $a_{0}a_{l}\neq 0$, then

[TABLE]

Remark 2.4*.*

Besser and Deninger defined the purely $p$-adic Mahler measure with use of the Shnirel’man integral and the $p$-adic log, which is different from ours, and proved an analogue of Jensen’s formula for it ([BD99]). We recall it in Section 4.

For our $\textsc{m}_{p}(f(t))$, we can remove the condition ${\rm gcd}(n,p)=1$ on the limit. In addition, we can extend this notion for $f(t)$ with roots on $|z|_{p}=1$. (We discuss another modification in Subsection 2.4.) Namely,

Theorem 2.5**.**

Let $f(t)\in{\mathbb{C}}_{p}[t^{\pm 1}]$ ($\neq 0$) which may have roots on $|z|_{p}=1$. If we modify the definition of the $p$-adic Mahler measure (Definition 2.3) as

[TABLE]

then Jensen’s formula (Theorem 2.3) still holds.

We will prove Theorem 2.5 in Subsection 2.3. It is nontrivial if $f(t)$ has a root on $|z|_{p}=1$ which is not a root of unity. (We have $|1+p|_{p}=1$ for instance.) Since there is no analogous notions of $\arg\alpha$ and the geometric trigonometric functions for $p$-adic numbers, we cannot directly follow the proof in [GAS91]. We need to give an alternative framework using properties of roots of unity and the strong triangle inequality.

By Theorem 2.5, we obtain a $p$-adic analogue of the asymptotic formula:

Proposition 2.6**.**

If $0\neq f(t)\in{\mathbb{C}}_{p}[t]$, then

[TABLE]

We note that the $p$-adic Mahler measure of a polynomial can be calculated easily from its coefficients (e.g. the argument in the end of [LW88]):

Proposition 2.7**.**

If $\displaystyle f(t)=\sum_{0\leq i\leq l}a_{i}t^{d-i}\in{\mathbb{C}}_{p}[t^{\pm 1}]$ with $d\in{\mathbb{Z}}$ and $l\in{\mathbb{N}}$, then $\textsc{m}_{p}(f(t))={\rm max}\{|a_{i}|_{p}\}_{i}$ holds. In other words, the $p$-adic Mahler measure and the Gauss norm of $f(t)$ coincide.

Proof.

Let the roots of $f(t)$ be indexed by natural numbers as $\alpha_{i}$ in a descending order with respect to the $p$-adic norm. If $r$ is the largest $i$ with $|\alpha_{i}|_{p}>1$, then we have $|a_{r}/a_{0}|_{p}=|\alpha_{1}\cdots\alpha_{r}+\text{smaller terms}|_{p}=|\alpha_{1}\cdots\alpha_{r}|_{p}$. ∎

2.3. Proofs

To begin with, we recall properties of the $p$-adic norm $|\cdot|_{p}$. It defines an ultra-metric, that is, it satisfies the strong triangle inequality $|x-y|_{p}\leq{\rm max}\{|x|_{p},|y|_{p}\}$ for every $x,y\in{\mathbb{C}}_{p}$. As a general property of ultra-metric spaces, we have the Krull sharpening: If $|x|_{p}\neq|y|_{p}$, then $|x-y|_{p}={\rm max}\{|x|_{p},|y|_{p}\}$ holds.

Next, we recall several properties of numbers on the $p$-adic unit circle.

Lemma 2.8**.**

If $\alpha\in{\mathbb{C}}_{p}$ satisfies $|\alpha|_{p}=1$ and $|\alpha-1|_{p}<1$, and if $m$ is prime to $p$, then $|\alpha^{m}-1|_{p}=|\alpha-1|_{p}$ holds.

Proof.

Indeed, we have $|\alpha^{m}-1|_{p}=|\alpha-1|_{p}|\alpha^{m-1}+\cdots+1|_{p}=|\alpha-1|_{p}|(\alpha^{m-1}-1)+\cdots+(\alpha-1)+m|_{p}=|\alpha-1|_{p}|(\alpha-1)(\alpha^{m-2}+\cdots+1)+m|_{p}$. Since $|\alpha-1|_{p}<1$, $|\alpha^{m-2}+\cdots+1|_{p}\leq{\rm max}\{|\alpha^{i}|_{p}\}_{i}=1$, and $|m|_{p}=1$, we have $|(\alpha-1)(\alpha^{m-2}+\cdots+1)+m|_{p}=1$. ∎

Lemma 2.9**.**

Let $\zeta$ be a primitive $n$-th root of unity and suppose $n=mp^{r}$ with $p\not|m$. If $n=1$, then $|1-\zeta|_{p}=0$. If $m=1$ and $r>0$, that is, if $\zeta$ is a primitive $p$-power-th root of unity with $\zeta\neq 1$, then $|1-\zeta|_{p}=p^{-1/(p-1)p^{r-1}}$ holds. If otherwise, then $|1-\zeta|_{p}=1$.

Proof.

We have $\displaystyle\prod_{\zeta^{n}=1,\zeta\neq 1}|1-\zeta|_{p}=|R(\nu_{n}(t),t-1)|_{p}=|\nu_{n}(1)|_{p}=|n|_{p}=|p^{r}|_{p}=p^{-r}$ and $|1-\zeta|_{p}\leq 1$. Since the set of primitive $p^{r}$-th roots of unity is closed by non-$p$ powers, the assertion follows from Lemma 2.8. ∎

Lemma 2.10**.**

If $\alpha\in{\mathbb{C}}_{p}$ satisfies $|\alpha|_{p}=1$, then there exist a unique $N$ with $p\not|N$ and a unique primitive $N$-th root $\xi$ of unity satisfying $|\alpha-\xi|_{p}<1$.

Proof.

We consider the valuation ring ${\mathcal{O}}_{p}=\{z\in{\mathbb{C}}_{p}\mid|z|_{p}\leq 1\}$ and its maximal ideal ${\mathfrak{m}}_{p}=\cup_{n\in{\mathbb{N}}_{>0}}(p^{1/n})$. Since $|\alpha|_{p}=1$, we have $\alpha\in{\mathcal{O}}_{p}$. We consider $\mathop{\mathrm{mod}}\nolimits{\mathfrak{m}}_{p}:{\mathcal{O}}_{p}\twoheadrightarrow\overline{{\mathbb{F}}}_{p}$. Elements of $\overline{{\mathbb{F}}}_{p}^{\times}$ are non-$p$-power-th roots of unity, and the restriction map \mathop{\mathrm{mod}}\nolimits{\mathfrak{m}}_{p}:\{z\in{\mathbb{C}}_{p}\mid\text{non-p-power-th roots of unity}\}\subset{\mathcal{O}}_{p}\to\overline{{\mathbb{F}}}_{p}^{\times} is a bijection.

Thus there exists a unique non-$p$-power-th root of unity $\xi\in{\mathbb{C}}_{p}$ satisfying $\xi\mathop{\mathrm{mod}}\nolimits{\mathfrak{m}}_{p}=\alpha\mathop{\mathrm{mod}}\nolimits{\mathfrak{m}}_{p}$, namely, $|\alpha-\xi|_{p}<1$. (This $\xi$ is the image of $\alpha\mathop{\mathrm{mod}}\nolimits{\mathfrak{m}}_{p}$ under the unique Teichmüller lift $\omega:\overline{{\mathbb{F}}}_{p}\hookrightarrow{\mathbb{C}}_{p}$.) ∎

Lemma 2.11**.**

If $\alpha\in{\mathbb{C}}_{p}$ satisfies $|\alpha|_{p}=1$ and $\alpha\neq 1$, then $\displaystyle\lim_{n\in{\mathbb{N}};\alpha^{n}\neq 1}|\alpha^{n}-1|_{p}^{1/n}=1$ holds.

Proof.

By Lemma 2.10, we have a unique non-$p$-power-th root $\xi$ of unity with $|\alpha-\xi|_{p}<1$. Suppose that $\xi$ is a primitive $N$-th root of unity. Then $N\neq 1$. Suppose $\alpha^{n}\neq 1$. Note that $|\alpha^{n}-1|_{p}=\prod_{\zeta^{n}=1}|\alpha-\zeta|_{p}=\prod_{\zeta^{n}=1}|(\alpha-\xi)+(\xi-\zeta)|_{p}$ holds.

If $N\not|n$ and $\zeta^{n}=1$, then $|\xi-\zeta|_{p}=|1-\zeta/\xi|_{p}=1$. Hence for any $n\in{\mathbb{N}}$ with $\alpha^{n}\neq 1$, we have $|\alpha^{n}-1|=1$. If $N|n$, then we have $\{|\xi-\zeta|_{p}\mid\zeta^{n}=1\}=\{|1-\zeta|_{p}\mid\zeta^{n}=1\}$. The product of all the non-zero elements of this set is $|n|_{p}$. By Lemma 2.9, if $\zeta$ runs through primitive $p^{r}$-th roots of unity for $r\in{\mathbb{N}}$, then $|1-\zeta|_{p}$ increase with respect to $r$ and converges to 1 as $r\to\infty$. If $\zeta$ is another root of unity with $\zeta\neq 1$, then $|1-\zeta|_{p}=1$. Hence we have $|\alpha-\zeta|_{p}=|\xi-\zeta|_{p}$ for almost all the roots $\zeta$ of unity.

Thus we have $|\alpha^{n}-1|_{p}=\prod_{\zeta^{n}=1}|\alpha-\zeta|_{p}=\bullet\times|n|_{p}$, where $\bullet$ lies in a finite set with $\bullet\neq 0$. Since $\displaystyle\lim_{n\in{\mathbb{N}}}|n|_{p}^{1/n}=1$, we obtain $\displaystyle\lim_{n\in{\mathbb{N}};\alpha^{n}\neq 1}|\alpha^{n}-1|_{p}^{1/n}=1$. ∎

The following lemma plays a key roll in the proof of Theorem 2.5:

Lemma 2.12**.**

For any element $\alpha\in{\mathbb{C}}_{p}$, the equality $\displaystyle\lim_{n\in{\mathbb{N}}}\prod_{\zeta^{n}=1;f(\zeta)\neq 0}|\zeta-\alpha|_{p}^{1/n}={\rm max}\{|\alpha|_{p},1\}$ holds, where $\zeta$ runs through roots of unity in ${\mathbb{C}}_{p}$.

Proof.

Suppose that $\alpha$ is not a root of unity. Then we have

[TABLE]

The denominator lies in a finite set independent of $n$. Hence $\text{(denominator)}^{1/n}\to 1$ as $n\to\infty$. If $|\alpha|_{p}\neq 1$, then $\text{(numerator)}^{1/n}=|\alpha^{n}-1|_{p}^{1/n}={\rm max}\{|\alpha|_{p}^{n},1\}^{1/n}={\rm max}\{|\alpha|_{p},1\}$ holds. If $|\alpha|_{p}=1$, then Lemma 2.11 assures $\displaystyle\lim_{n}|\alpha^{n}-1|_{p}^{1/n}=1$.

Suppose instead that $\alpha$ is a root of unity. Then we have

[TABLE]

We have $\text{(denominator)}^{1/n}\to 1$ as $n\to\infty$. For $n$’s with $\alpha^{n}\neq 1$, we have $\text{(numerator)}=|\alpha^{n}-1|_{p}$. The previous lemma assures $\text{(numerator)}^{1/n}\to 1$ as $n\to\infty$. For $n$’s with $\alpha^{n}=1$, we have $\displaystyle\text{(numerator)}=\prod_{\zeta^{n}=1,\zeta\neq 1}|\zeta-1|_{p}=|n|_{p}$. Since $1/n\leq|n|_{p}\leq 1$, we have $\text{(numerator)}^{1/n}\to 1$ as $n\to\infty$. ∎

Proof of Theorem 2.5.

Let $\displaystyle f(t)=a_{0}t^{d-l}\prod_{i}(t-\alpha_{i})\in{\mathbb{C}}_{p}[t^{\pm 1}]$. It is sufficient to prove

[TABLE]

where $\zeta$ runs through roots of unity in ${\mathbb{C}}_{p}$. Since the both side of this equality is multiplicative with respect to $f$, it is sufficient to prove the assertion on each factor $t-\alpha_{i}$. Now the assertion follows from Lemma 2.12. ∎

2.4. Alternative modification of the integral path

The Euclidean Mahler measure of $f(t)\in{\mathbb{C}}[t^{\pm 1}]$ satisfying Jensen’s formula was defined as the limit of the integral along a path $\gamma:S^{1}\to{\mathbb{C}}$ as $\gamma$ uniformly approaches the unit circle ([EW99]).

Here we give an analogous argument for our $p$-adic Mahler measure of $f(t)\in{\mathbb{C}}_{p}[t^{\pm 1}]$. We consider the image of roots of unity under a function $g:{\mathbb{C}}_{p}\to{\mathbb{C}}_{p}$ and let $g$ approach ${\rm id}$ uniformly with respect to the $p$-adic topology. Namely, for each function $g$ with $f(g(\zeta))\neq 0$ for any root $\zeta$ of unity, we put $\displaystyle\log\textsc{m}_{p,g}(f(t)):=\lim_{n\in{\mathbb{N}}}\frac{1}{n}\sum_{\zeta^{n}=1}\log|f(g(\zeta))|_{p}$, where the limit is taken with respect to the Euclidean topology. For each $\varepsilon>0$, we consider $g$’s with ${\rm sup}\{|g(\zeta)-\zeta|_{p}\mid\zeta\in\text{(roots of unity)}\}<\varepsilon$ and let $\varepsilon\to 0$. (Such $g$ exists for each $\varepsilon$, e.g., $g(z)=z+\varepsilon/2$.) Then we have

Proposition 2.13**.**

For any function $g$ sufficiently close to ${\rm id}$, the limit $\textsc{m}_{p,g}(f(t))$ exists and the value is independent of $g$. The limit $\displaystyle\lim_{g\to{\rm id}}\textsc{m}_{p}(f(g(t)))$ exists and coincides with $\textsc{m}_{p}(f(t))$ satisfying Jensen’s formula.

Proof.

Since $\log|\bullet|_{p}$ satisfies the logarithm laws, it is sufficient to consider $f(t)=t-\alpha$ for $\alpha\in{\mathbb{C}}_{p}$. If $|\alpha|_{p}\neq 1$, then the assertion immediately follows from a property of ultra-metric. If $|\alpha|_{p}=1$, then we have a unique non-$p$-power-th root $\xi$ of unity satisfying $|\alpha-\xi|_{p}<1$. Assume ${\rm sup}\{|g(\zeta)-\zeta|_{p}\mid\zeta\in\text{(roots of unity)}\}<\varepsilon<1$. If $\zeta^{n}=1$, then we have $|g(\zeta)-\alpha|_{p}=|(g(\zeta)-\zeta)+(\zeta-\xi)+(\xi-\alpha)|_{p}$. By a similar argument to Lemma 2.11, we obtain $\displaystyle\lim_{n\in{\mathbb{N}}}\prod_{\zeta^{n}=1}|g(\zeta)-\alpha|_{p}=1$. Therefore we have $\textsc{m}_{p,g}(f(t))={\rm max}\{|\alpha|_{p},1\}$. ∎

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

3.1. ${\mathbb{Z}}$-covers and $\textsc{m}_{p}$

Let $M$ be a ${\mathbb{Q}}$HS3 and let $L$ be a $d$-component link in $M$. Let $h_{\infty}:X_{\infty}\to X=M-L$ be a ${\mathbb{Z}}$-cover, which we call a ${\mathbb{Z}}$-cover over $(M,L)$, with a generator $t$ of the deck transformation group $t^{{\mathbb{Z}}}$. Then the Alexander polynomial $A_{L}(t)\in{\mathbb{Z}}[t]\subset{\mathbb{Z}}[t^{\mathbb{Z}}]$ is a generator of the principal ideal of the Fitting ideal ${\rm Fitt}_{{\mathbb{Z}}[t^{\mathbb{Z}}]}H_{1}(X_{\infty})$ of the Alexander module over ${\mathbb{Z}}[t^{\mathbb{Z}}]$, as explained in Section 1.

Example 3.1**.**

Suppose that every component $L_{i}$ of $L$ is null-homologous, and let $t_{i}\in H_{1}(X)$ denote the meridian of each $L_{i}$. Then a standard ${\mathbb{Z}}$-cover called the total linking number cover (TLN-cover) over $(M,L)$ is defined by the surjective homomorphism $\tau:H_{1}(X,{\mathbb{Z}})\twoheadrightarrow{\mathbb{Z}};$ $\forall t_{i}\mapsto 1$. If $A_{L}(t)\neq 0$, then we have ${\rm Fitt}_{{\mathbb{Z}}[t^{\mathbb{Z}}]}H_{1}(X_{\infty})=(A_{L}(t))$ by [KM08, Lemma 3.1].

If $M=S^{3}$, then the multivariable Alexander polynomial $\Delta_{L}(t_{1},\ldots,t_{d})$, the reduced Alexander polynomial $\Delta_{L}(t)=\Delta_{L}(t,\ldots,t)$, and the Hosokawa polynomial $H_{L}(t)$ of $L$ are defined. For the TLN-cover over $(S^{3},L)$, we have

[TABLE]

For each $n\in{\mathbb{N}}_{>0}$, let $M_{n}\to M$ denote the branched ${\mathbb{Z}}/n{\mathbb{Z}}$-cover obtained as the Fox completion of the ${\mathbb{Z}}/n{\mathbb{Z}}$-subcover of $h_{\infty}$. Then the orders of groups and the cyclic resultants of $A_{L}(t)$ satisfy the following well-known formula:

Proposition 3.2**.**

Let the notation be as above. Then there exists some finite set ${\mathcal{C}}$ of ${\mathbb{N}}$ such that for any $n\in{\mathbb{N}}_{>0}$

[TABLE]

holds for some $C\in{\mathcal{C}}$.

This proposition follows from Sakuma’s theorem ([Sak81, Theorem 3]) or Porti’s theorem ([Por04, Theorem 1.1]) by a similar argument to [KM08, Proof of Theorem 2.1]. It was initially proved for a knot $L$ in $M=S^{3}$ by Fox ([Fox56], [Web79]) and for a link $L$ in $M=S^{3}$ by Mayberry–Murasugi ([MM82]).

By Propositions 2.1, 2.6, and Theorem 3.2, we obtain the following asymptotic formulae:

Theorem 3.3**.**

Let the notation be as above. If $A_{L}(t)\neq 0$, then

[TABLE]

The Euclidean formula for knots in $S^{3}$ was proved by [GAS91], [Ril90]. Riley also gave an estimation of $||H_{1}(M_{n})||_{p}^{-1}$ from above.

We denote $p=\infty$ for Euclidean objects. For $f(t)\in{\mathbb{Z}}[t^{\pm 1}]$, the set of $p$-adic Mahler measures $\textsc{m}_{p}(f(t))$ for $p<\infty$ do not tell that for $p=\infty$. The set of those for $p\leq\infty$ would have some meaning.

Example 3.4**.**

We use the notation in the Rolfsen table ([Rol76]).

(1) If $K=4_{1}$ (figure 8-knot), then the Alexander polynomial $\Delta_{K}(t)=t^{2}-3t+1$ satisfies $\textsc{m}(\Delta_{K}(t))=\frac{3+\sqrt{5}}{2}$, $\textsc{m}_{p}(\Delta_{K}(t))=0$ ($p<\infty$).

(2) If $L=4^{2}_{1}$ (Solomon’s knot, sigillum Salomonis), then one of the reduced Alexander polynomial $\Delta_{L}(t,t^{-1})=2$ satisfies $\textsc{m}(2)=0,\textsc{m}_{2}(2)=1/2,\textsc{m}_{p}(2)=0$ ($p\neq 2$). The same values are obtained for $A_{L}(t)=2(t-1)$.

Remark 3.5*.*

(${\mathbb{Z}}^{d}$-covers, non-${\mathbb{Q}}$HS3-cases, etc.) We have an asymptotic formula for ${\mathbb{Z}}^{d}$-covers with use of the multivariable Alexander polynomials and the Mahler measures. In addition, even if $M_{n}$’s are not necessarily ${\mathbb{Q}}$HS3, we have an asymptotic formula of the torsion subgroups $H_{1}(M_{n})_{\rm tor}$ with use of the higher Alexander polynomials and the Mahler measures ([SW02], [Le14]).

It is noteworthy that sometimes we obtain special values of zeta functions, $L$-functions, and multiple zeta functions as the Mahler measures of multivariable polynomials ([Smy81], [Lal03]). Furthermore, in several examples the hyperbolic volumes of knot exteriors relate to the Mahler measures of the $A$-polynomials ([Boy02], [BRVD03], [Lal04]). An asymptotic formula with use of the hyperbolic volumes is known as the Bergeron–Venkatesh conjecture and Le’s theorem ([BV13], [Le14]). We expect $p$-adic analogues of them.

Finally we remark that a recent development of such a study for twisted Alexander polynomials of knots is due to Tange ([Tan17]).

3.2. Iwasawa $\mu_{p}$-invariant and $\textsc{m}_{p}$

Let $M$ be a ${\mathbb{Q}}$HS3, let $L$ be a link in $M$, and let $\{h_{p^{r}}:M_{p^{r}}\to M\}_{r}$ be an inverse system of branched ${\mathbb{Z}}/p^{r}{\mathbb{Z}}$-covers over $(M,L)$. (For such an object, see Subsection 5.1 also.) Let $t$ denote the topological generator of the inverse limit of the deck transformation groups ${\mathbb{Z}}/p^{r}{\mathbb{Z}}$ corresponding to 1. Then the Iwasawa module ${\mathcal{H}}_{p}:=\varprojlim_{r}H_{1}(M_{p^{r}},{\mathbb{Z}}_{p})$ is a finitely generated module over ${\mathbb{Z}}_{p}[[t^{{\mathbb{Z}}_{p}}]]\cong{\mathbb{Z}}_{p}[[T]];t\mapsto 1+T$. By [Uek17, Theorem 4.17], $M_{p^{r}}$’s are ${\mathbb{Q}}$HS3’s if and only if its characteristic polynomial in ${\mathbb{Z}}_{p}[[t^{{\mathbb{Z}}_{p}}]]$ does not vanish at any $p$-power-th root of unity.

The following formula is an analogue of Iwasawa’s class number formula ([Iwa59]), which was initially given by [HMM06] and generalized by [KM08] and [Uek17]:

Proposition 3.6** (Iwasawa type formula).**

Let the notation be as above. If $M_{p^{n}}$’s are ${\mathbb{Q}}$HS3’s, then there exist some $\lambda_{p},\mu_{p}\in{\mathbb{N}},\nu_{p}\in{\mathbb{Z}}$ called the Iwasawa invariants such that for any $r\gg 0$ in ${\mathbb{N}}$

[TABLE]

holds.

Suppose that the ${\mathbb{Z}}/p^{r}{\mathbb{Z}}$-covers are obtained from a ${\mathbb{Z}}$-cover over $(M,L)$ with the Alexander polynomial $A_{L}(t)$. Comparing with the results in the previous section, we obtain

Proposition 3.7**.**

[TABLE]

In this sense, the Iwasawa type formula is a $p$-adic refinement of the asymptotic formula (Theorem 3.3) with use of $\textsc{m}_{p}$.

Remark 3.8*.*

We review what we have done for ${\mathbb{Z}}$-covers of $S^{3}$ branched over links $L$. By the Mayberry–Murasugi formula and the definition of $\textsc{m}_{p}$ imitating the Shnirel’man integral, if $A_{L}(t)$ does not vanish on the $p$-adic unit circle, then we have $\displaystyle\lim_{n\in{\mathbb{N}};\gcd(p,n)=1}||H_{1}(M_{n})||_{p}=\textsc{m}_{p}(A_{L}(t))$. By the Iwasawa type formula, if $A_{L}(t)$ does not vanish at any $p$-power-th root of unity, then we have $\displaystyle\lim_{n=p^{r}}||H_{1}(M_{n})||_{p}=p^{-\mu_{p}}$. Note that the intersection of their domains of $n$ is empty. By Jensen’s formula, the formula $\textsc{m}_{p}(A_{L}(t))={\rm max}\{|\text{coefficients}|_{p}\}$, and the $p$-adic Weierstrass preparation theorem, if $A_{L}(t)$ has no root on $|z|_{p}=1$, then $\log\textsc{m}_{p}=-\mu_{p}\log p$ immediately follows. We removed the assumption on $A_{L}(t)$ by extending the definition of $\textsc{m}_{p}$.

We have $\textsc{m}_{p}(A_{L}(t))={\rm max}\{|\text{coefficients}|_{p}\}$. If $L=K$ is a knot, then $A_{L}(t)=\Delta_{K}(t)$ satisfies $\Delta_{K}(1)=\pm 1$ and its is primitive. Thus for each $n<\infty$, we have $\textsc{m}_{p}(A_{L}(t))=1$ and $\mu_{p}=0$. In regard to $d\geq 2$-component link $L$, various examples with $\textsc{m}_{p}(A_{L}(t))<1$ and $\mu_{p}>0$ are given ([KM08], [KM13], [Uek16]).

Example 3.9**.**

We list up all the examples of TLN-covers with non-trivial $\mu_{p}$ obtained from 2-component links $L$ in the Rolfsen table ([Rol76]). We have $A_{L}(t)=(t-1)\Delta_{L}(t^{\epsilon_{1}},t^{\epsilon_{2}}),\epsilon_{i}\in\{\pm 1\}$. We denote $f\dot{=}g$ if $f$ and $g$ coincide up to multiplication by units of ${\mathbb{Z}}[t^{\pm 1}]$.

(1) $L=4^{2}_{1}$. $\Delta_{L}(x,y)=1+xy$, $\Delta_{L}(t,t^{-1})=2$, $\mu_{2}=1$.

(2) $L=6^{2}_{1}$. $\Delta_{L}(x,y)=1+xy+x^{2}y^{2}$, $\Delta_{L}(t,t^{-1})=3$, $\mu_{3}=1$.

(3) $L=6^{2}_{3}$. $\Delta_{L}(x,y)=2-x-y+2xy$, $\Delta_{L}(t,t)=2(t^{2}-t+1)$, $\mu_{2}=1$.

(4) $L=7^{2}_{3}$. $\Delta_{L}(x,y)=1-x-y+xy$, $\Delta_{L}(t,t)\,$ $\dot{=}\,\Delta_{L}(t,t^{-1})\,\dot{=}\,4(t^{2}-1)^{2}$, $\mu_{2}=2$.

3.3. $p$-adic entropy $h_{p}$ and $\textsc{m}_{p}$

Noguchi studied ${\mathbb{Z}}$-covers of $S^{3}$ branched over knots ([Nog07]). Let $t$ denote the meridian action on the (${\mathbb{Q}}$-)Alexander module $H_{1}(X_{\infty},{\mathbb{Q}})\cong{\mathbb{Q}}^{l}$, and let ${\widehat{t}}$ denote the solenoidal system obtained as the Pontrjagin dual of $t$. He applied the result of Lind and Ward ([LW88]) to ${\widehat{t}}$ and calculated the topological entropy $h_{\rm top}$ of ${\widehat{t}}$, which coincides with Bowen’s measure theoretic entropy.

Giordano-Bruno and Virili ([GBV15]) studied the algebraic entropies $h_{\rm alg}$ of dynamical systems of ${\mathbb{Q}}$-linear spaces. They gave a formula parallel to that of [LW88], so that $h_{\rm alg}$ of $t$ coincides with $h_{\rm top}$ of ${\widehat{t}}$ above. In what follows, we just call them the entropy.

These entropies have their origin in the Ergodic theory. They were introduced by Adler, Konheim, and McAndrew ([AKM65]). Various generalization and duality have been studied ([DGB14] for instance).

Results of [LW88] and [GBV15] called the Yuzvinski formula or the Kolmogorov–Sinai formula can be stated with use of our $p$-adic Mahler measure as follows:

Proposition 3.10**.**

Let $\varphi\in\mathop{\mathrm{GL}}\nolimits(l,{\mathbb{Q}})$ with $l\in{\mathbb{N}}$ and let $B(t)\in{\mathbb{Q}}[t]$ denote its (monic) characteristic polynomial. Then the algebraic entropy of $\varphi\curvearrowright{\mathbb{Q}}^{l}$ and the topological entropy of its Pontrjagin dual are given by

[TABLE]

where $h_{p}$ denotes the $p$-adic entropy of $\varphi\curvearrowright{\mathbb{Q}}^{l}\otimes{\mathbb{Q}}_{p}\cong{\mathbb{Q}}_{p}^{l}$ or its Pontrjagin dual for $p\leq\infty$.

In addition, suppose $\displaystyle B(t)=\sum_{0\leq i\leq l}b_{i}t^{d-i}\in{\mathbb{Q}}[t]$ with $d,l\in{\mathbb{N}}$. Then $\displaystyle\sum_{p<\infty}h_{p}=\log s$ hols for $s={\rm lcm}\{b_{i}\}_{i}$, and the primitive polynomial $sB(t)\in{\mathbb{Z}}[t]$ satisfies $h=\log\textsc{m}(sB(t))$.

Now let $L$ be a link in $S^{3}$ and let $h_{\infty}:X_{\infty}\to X=S^{3}-L$ be a TLN-${\mathbb{Z}}$-cover with the Alexander polynomial $\displaystyle A_{L}(t)=\sum_{0\leq i\leq l}a_{i}t^{d-i}\in{\mathbb{Z}}[t]$ with $a_{0}a_{l}\neq 0$. Then we have $H_{1}(X_{\infty};{\mathbb{Q}})\cong{\mathbb{Q}}^{l}$ as linear spaces over ${\mathbb{Q}}$ and the characteristic polynomial of the meridian action on $H_{1}(X_{\infty};{\mathbb{Q}})$ is given by the associated monic polynomial $A_{L}(t)/a_{0}$. Hence we have

Proposition 3.11**.**

The entropy of the $t$ action on $H_{1}(X_{\infty};{\mathbb{Q}})$ is given by

[TABLE]

By the product formula $\displaystyle\prod_{p\leq\infty}|a_{0}|_{p}=1$, we also have $\displaystyle h=\sum_{p\leq\infty}\log\textsc{m}_{p}(A_{L}(t))$. In addition, the primitive polynomial $A_{L}(t)/{\rm gcd}\{a_{i}\}_{i}\in{\mathbb{Z}}[t]$ satisfies the equality $h=\textsc{m}(A_{L}(t)/{\rm gcd}\{a_{i}\}_{i})$.

If $L$ is a knot, then $A_{L}(t)=\Delta_{L}(t)$ is primitive, and similar results hold for the Alexander module $H_{1}(X_{\infty})$ over ${\mathbb{Z}}$ ([Nog07, Theorem 1]). If $L$ is a general link, then $A_{L}(t)$ is not necessarily primitive. The entropy $h_{\mathbb{Z}}$ of the meridian action on $H_{1}(X_{\infty})$ over ${\mathbb{Z}}$ is given by $h_{\mathbb{Z}}=\log\textsc{m}(A_{L}(t))$ and satisfies $h_{\mathbb{Z}}=h+\log{\rm gcd}\{a_{i}\}_{i}$.

Example 3.12**.**

In the Rolfsen table ([Rol76]), the first example with nontrivial entropy $h$ is $K=4_{1}$ (figure 8-knot). We have $\Delta_{K}(t)=t^{2}-3t+1$, $h=\log\textsc{m}_{\infty}(\Delta_{K}(t))=\log\frac{3+\sqrt{5}}{2}$. We list up all the 2-component links $L$ with nontrivial $h$ in the table:

(1) $L=6^{2}_{2}$. $\Delta_{L}(x,y)=x+y-xy+x^{2}y+xy^{2}$, $\Delta_{L}(t,t)\,\dot{=}\,\Delta_{L}(t,t^{-1})\,\dot{=}\,2t^{2}-t+2$, $h=h_{2}=\log 2$.

(2) $L=6^{2}_{3}$. $\Delta_{L}(x,y)=2-x-y+2xy$, $\Delta_{L}(t,t^{-1})\,\dot{=}\,t^{2}-4t+1$, $h=h_{\infty}=\log(2+\sqrt{3})$.

(3) $L=7^{2}_{1}$. $\Delta_{L}(x,y)=1-x-y+xy-x^{2}y-xy^{2}+x^{2}y^{2}$. (i) $\Delta_{L}(t,t)=t^{4}-2t^{3}+t^{2}-2t+1$, $h=h_{\infty}=\log\frac{1+\sqrt{2}+\sqrt{2\sqrt{2}-1}}{2}$. (ii) $\Delta_{L}(t,t^{-1})\,\dot{=}\,2t^{2}-3t+2$, $h=h_{2}=\log 2$.

(4) $L=7^{2}_{2}$. $\Delta_{L}(x,y)=1-x-y+3xy-x^{2}y-xy^{2}+x^{2}y^{2}$, $\Delta_{L}(t,t^{-1})\,\dot{=}\,2t^{2}-5t+2=2(t-2)(t-\frac{1}{2})$, $h_{\infty}=2$, $h_{2}=\log|1/2|_{2}=\log 2$, $h=h_{\infty}+h_{2}=\log\textsc{m}(2t^{2}-5t+2)=2\log 2$.

3.4. $h_{p}$, $\mu_{p}$, and the leading coefficient $a_{0}$

We continue to study a TLN-${\mathbb{Z}}$-cover of $(S^{3},L)$. The leading coefficient $a_{0}$ of $A_{L}(t)$ has geometric information. For instance, if $L$ is fibered, then $a_{0}=1$ holds (e.g., [Hil12, Theorem 5.12]).

Now let $p$ be a prime number again. Then $h_{p}=\log\textsc{m}_{p}(A_{L}(t)/a_{0})$ measures the ratio between the monic and primitive polynomials associated to $A_{L}(t)$, while $\textsc{m}_{p}(A_{L}(t))=p^{-\mu_{p}}$ measures the ratio between $A_{L}(t)$ itself and the primitive one.

Since $\textsc{m}_{p}(A_{L}(t)/a_{0})=\textsc{m}_{p}(A_{L}(t))/|a_{0}|_{p}$, we obtain a balance formula among $h_{p},\mu_{p},$ and $a_{0}$:

Proposition 3.13**.**

[TABLE]

Example 3.14**.**

We have only one example of 2-component link whose $h_{p},\mu_{p}$ are both nontrivial in the Rolfsen table ([Rol76]). It is $L=9^{2}_{23}$. We have $\Delta_{L}(x,y)=x^{2}-2xy+y^{2}x^{3}-4x^{2}y-4xy-y^{3}+x^{3}y+2x^{2}y^{2}+xy^{3}$, $\Delta_{L}(t,t)=4t^{2}-10t+4=2(2t^{2}-5t+2)=2(2-t)(1-2t)$. $|a_{0}|_{2}=1/4$. $h_{\infty}=\log 2$, $h_{2}=\log|1/2|_{2}=\log 2$, $h=h_{\infty}+h_{2}=\log\textsc{m}_{\infty}(2t^{2}-5t+2)=2\log 2$. $\textsc{m}_{2}(\Delta_{L}(t,t))=|2|_{2}=1/2$, $\mu_{2}=1$.

If $A_{L}(t)=a_{0}\prod_{i}(t-\alpha_{i})$, then by Theorems 2.3, 2.5, and Proposition 3.11, we have $\displaystyle h_{p}=\sum_{|\alpha_{i}|_{p}>1}\log|\alpha_{i}|_{p}$. Note the product formula $\displaystyle\displaystyle\prod_{p<\infty}|a_{0}|_{p}=1/|a_{0}|$. Then the previous proposition gives

Corollary 3.15**.**

[TABLE]

This formula generalizes [Nog07, Corollary 4]. Indeed, if $L$ is a knot, then $\mu_{p}=0$ for any $p<\infty$, and hence $\displaystyle\log|a_{0}|=\sum_{p<\infty}\sum_{|\alpha_{i}|_{p}>1}\log|\alpha_{i}|_{p}$ $\displaystyle(=\sum_{p<\infty}h_{p})$ holds.

4. Purely $p$-adic notions

4.1. Besser–Deninger $m_{p}$

The notion of the $p$-adic logarithm $\log_{p}:{\mathbb{C}}_{p}^{*}\to{\mathbb{C}}_{p}$ is known ([Was97, Chapter 5]). It is first defined on the unit disc $|z-1|_{p}<1$ by the power series $\displaystyle\log_{p}(1+z)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}z^{n}}{n}$, and then uniquely extended to whole ${\mathbb{C}}_{p}^{*}$ by the $p$-adic analytic continuation under the normalization $\log_{p}p=0$. It is sometimes called the Iwasawa $p$-adic logarithm. Besser and Deninger introduced the purely $p$-adic logarithmic Mahler measure $m_{p}$ for $p$-adic analytic functions ([BD99]). It is different from ours, and is defined by the Shnirel’man integral ([Shn38]):

Definition 4.1**.**

For a $p$-adic analytic functions $f(z)$ with no zero on $|z|_{p}=1$, we put

[TABLE]

where the limit is taken with respect to the $p$-adic topology.

It satisfies Jensen’s formula for Laurent polynomials:

Proposition 4.2** (Jensen’s formula).**

If $f(t)\in{\mathbb{C}}_{p}[t^{\pm 1}]$ with no root on $|z|_{p}=1$, then the limit in Definition 4.1 exists. If $\displaystyle f(t)=\sum_{0\leq i\leq l}a_{i}t^{d-i}=a_{0}t^{d-l}\prod_{0\leq i\leq l}(t-\alpha_{i})$ ($a_{i},\alpha_{i}\in{\mathbb{C}}_{p},a_{0}a_{l}\neq 0$), then we have

[TABLE]

Since $\log_{p}(z)$ does not vanish on $|z|_{p}=1$, we have no natural generalization of $m_{p}$ for $f(z)$ with zeros on $|z|_{p}=1$.

Remark 4.3*.*

In the Shnirel’man integral, we have no term corresponding to $2\pi\sqrt{-1}$ in the complex case.In [Mih12], its denormalization and generalization are given, in which the corresponding terms called periods in $\mathbb{B}_{\rm dR}$ appear.

By the definition of the Shnirel’man integral, an analogue of the asymptotic formula of resultants (Proposition 2.1, 2.6) immediately follows:

Proposition 4.4**.**

If $f(z)\in{\mathbb{C}}_{p}[t]$ with no root on $|z|_{p}=1$, then

[TABLE]

Since $R(f(t),t^{n}-1)=R(f(t),\nu_{n}(t))\times f(1)$, if $f(1)\neq 0$, then the convergence of $\{\displaystyle\frac{1}{n}\log_{p}|R(f(t),\nu_{n}(t))|\}_{n}$ is equivalent to that of $\{\displaystyle\frac{1}{n}\log_{p}|f(1)|\}_{n}$, and hence to $\log_{p}|f(1)|=0$.

We apply their theory to the TLN-${\mathbb{Z}}$-cover of $S^{3}$ branched over a $d$-component link $L$. Put $\Lambda={\mathbb{Z}}[t^{\pm 1}]$. We use the notation in the previous section. We have $A_{L}(t)=(t-1)^{d}H_{L}(t)$. The Alexander module admits a natural direct sum decomposition $H_{1}(X_{\infty})\cong{\mathbb{Z}}^{d-1}\oplus H_{1}^{\prime}(X_{\infty})$ of ${\mathbb{Z}}[t^{\mathbb{Z}}]$-modules with ${\rm Fitt}_{{\mathbb{Z}}[t^{\mathbb{Z}}]}H_{1}^{\prime}(X_{\infty})=(H_{L}(t))$.

If $d=1$, then $A_{L}(t)$, $H_{L}(t)$ and the Alexander polynomial $\Delta_{L}(t)$ coincide. Since $A_{L}(1)=\pm 1$, by Proposition 3.2, we have $|H_{1}(M_{n})|=|R(A_{L}(t),\nu_{n}(t)))|=|R(A_{L}(t),t^{n}-1)|$. We obtain an analogue of Theorem 3.3 with respect to the $p$-adic norm: If $A_{L}(t)$ has no root on $|z|_{p}=1$, then

[TABLE]

For the cases with $d>1$, note that the trivial action of $t$ on ${\mathbb{Z}}$ is not expansive and $\hbar_{p}$ is not defined. It comes from the fact that $\{(\log_{p}n)/n\}_{n}$ is not convergent in ${\mathbb{C}}_{p}$. However, for any $d$, if $H_{L}(z)$ has no roots on $|z|_{p}=1$, then the meridian action $t$ on the Hosokawa module $\Lambda/(H_{L}(t))$ is expansive, and its purely $p$-adic entropy is given by $\hbar_{p}=m_{p}(H_{L}(t))$.

For a general $d\in{\mathbb{N}}_{>0}$, we have $|H_{1}(M_{n})|=|R(A_{L}(t),\nu_{n}(t)))|=|R(H_{L}(t),\nu_{n}(t))$ $R((t-1)^{d-1},\nu_{n}(t))|=|R(H_{L}(t),t^{n}-1)/R(H_{L}(t),t-1)|\times n^{d-1}=|R(H_{L}(t),t^{n}-1)/H_{L}(1)|\times n^{d-1}.$ Hence a modified asymptotic formula for links is stated as follows:

Theorem 4.5**.**

Let $L$ be a $d$-component link in $S^{3}$ and suppose that the Hosokawa polynomial $H_{L}(t)$ has no root on $|z|_{p}=1$. Then the TLN-${\mathbb{Z}}$-cover over $(S^{3},L)$ satisfies

[TABLE]

with respect to the $p$-adic topology.

4.2. Deninger’s $\hbar_{p}$

The topological entropy of a dynamical system (an automorphism $\varphi$ on a space $X$) is sometimes calculated by using the numbers of the fixed points (e.g., [EW99, Section 2]):

[TABLE]

Deninger defined purely $p$-adic entropy $\hbar_{p}(\varphi)$ by using the numbers of the fixed points and $p$-adic logarithm under certain situations:

[TABLE]

A Laurent polynomial $f$ has no zeros on the $p$-adic unit torus if and only if the associated solenoidal dynamical system is expansive. Under those conditions, we have $\hbar_{p}=m_{p}(f)$ ([Den09, Theorem 1]).

We consider the TLN-${\mathbb{Z}}$-cover over a $d$-component link $L$ in $S^{3}$ again. We consider the meridian action on the Alexander module $H_{1}(X_{\infty})\cong{\mathbb{Z}}^{d-1}\oplus\Lambda/(H_{L}(t))$. Since $\{(\log_{p}n)/n\}_{n}$ is not convergent in ${\mathbb{C}}_{p}$, the trivial action of $t$ on ${\mathbb{Z}}$ is not expansive. Therefore, if $d>1$, then $\hbar_{p}$ is not defined for $t\curvearrowright H_{1}(X_{\infty})$. However, we have a formula for a direct summand:

Proposition 4.6**.**

For any $d$, if $H_{L}(z)$ has no root on $|z|_{p}=1$, then the meridian action on the Hosokawa module $H_{1}^{\prime}(X_{\infty})$ is expansive, and its purely $p$-adic entropy is given by $\hbar_{p}=m_{p}(H_{L}(t))$.

Example 4.7**.**

The first example of a knot in Rolfsen’s table for which $\hbar_{p}$ is defined is $L=5_{2}$ (the 3-twist knot). We have $A_{L}(t)=\Delta_{L}(t)=2t^{2}-3t+2$. Its roots $\alpha,\beta$ satisfy $|\alpha\beta|_{2}=1$ and $|\alpha+\beta|_{2}=|3/2|_{2}=2>1$. Let $\alpha=(3-\sqrt{-7})/4$ denote the larger root. Then we have $\hbar_{2}=m_{2}(\Delta_{L}(t))=\log_{2}(3-\sqrt{-7})$.

Among the examples of links we have seen, $\hbar_{p}$ is defined only for the following cases with $p=2$:

Example 3.12 (1) $L=6^{2}_{2}$, $H_{L}(t)\dot{=}2t^{2}-t+2$. Let $(1-\sqrt{-15})/4$ denote the larger root. Then we have $\hbar_{2}=m_{2}(A_{L}(t)/2)=\log_{2}(1-\sqrt{-15})$.

(3) $L=7^{2}_{1}$. (ii) $H_{L}(t)\dot{=}2t^{2}-3t+2$. Let $(3-\sqrt{-7})/4$ denote larger root. Then $\hbar_{2}=\log_{2}(3-\sqrt{-7})$.

(4) $L=7^{2}_{2}$, $H_{L}(t)\dot{=}2t^{2}-5t+2=2(t-2)(t-\frac{1}{2})$. $\hbar_{2}=0$.

5. Profinite cyclic covers

We study entropies of the meridian actions on modules associated to profinite cyclic covers.

5.1. ${\mathbb{Z}}_{p}$-covers

Let $L$ be a link in $S^{3}$. A branched ${\mathbb{Z}}_{p}$-cover over $(S^{3},L)$ is an inverse system $\{M_{p^{n}}\to S^{3}\}_{n}$ of branched ${\mathbb{Z}}/p^{n}{\mathbb{Z}}$-covers branched over $L$. If we fix a ${\mathbb{Z}}$-cover of $S^{3}-L$, then we obtain a branched ${\mathbb{Z}}_{p}$-cover from its subcovers. A branched ${\mathbb{Z}}_{p}$-cover is not necessarily obtained from a ${\mathbb{Z}}$-cover, because the image of ${\mathbb{Z}}^{2}\to{\mathbb{Z}}_{p}$ is not necessarily sent to ${\mathbb{Z}}\subset{\mathbb{Z}}_{p}$ by automorphisms of ${\mathbb{Z}}_{p}$.

We have an isomorphism ${\mathbb{Z}}_{p}[[t^{{\mathbb{Z}}_{p}}]]\cong{\mathbb{Z}}_{p}[[T]];t\mapsto 1+T$. The Iwasawa module (pro-$p$ Alexander module) ${\mathcal{H}}_{p}:=\varprojlim_{n}H_{1}(M_{p^{n}},{\mathbb{Z}}_{p})$ of a branched ${\mathbb{Z}}_{p}$-cover is a finitely generated torsion ${\mathbb{Z}}_{p}[[T]]$-module ([Uek17]). Let $\tau:H_{1}(S^{3}-L)\to{\mathbb{Z}}_{p};t_{i}\mapsto z_{i}$ ($z_{i}\in{\mathbb{Z}}_{p}^{\times}$) be a homomorphism. Then $\mathop{\mathrm{Ker}}\nolimits$ of the composite of $\tau$ and the natural surjections ${\mathbb{Z}}_{p}\twoheadrightarrow{\mathbb{Z}}/p^{n}{\mathbb{Z}}$ defines an inverse system of ${\mathbb{Z}}/p^{n}{\mathbb{Z}}$-covers, and hence a totally branched ${\mathbb{Z}}_{p}$-cover associated to $\tau$. The Fitting ideal of ${\mathcal{H}}_{p}$ over ${\mathbb{Z}}_{p}[[t^{{\mathbb{Z}}_{p}}]]$ is generated by $A_{L}(t)=(t-1)\Delta_{L}(t^{z_{1}},\ldots,t^{z_{d}})$.

By the $p$-adic Weierstrass preparation theorem [Was97, Theorem 7.3], we have $A_{L}(1+T)=p^{\mu_{p}}g(T)u(T)$ for a distinguished polynomial $g(T)$ (that is, a monic polynomial with every lower coefficient divisible by $p$) and $u(T)\in{\mathbb{Z}}_{p}[[T]]^{\times}$. All the roots of $g(T)$ are on $|z|_{p}<1$ and $\lambda_{p}=\deg(g(T))$ holds. If $u(T)$ has a zero on the domain of convergence, then it is on $|z|_{p}\geq 1$. We have $\displaystyle{\mathcal{H}}\otimes{\mathbb{Q}}\cong\bigoplus_{i}{\mathbb{Q}}_{p}[[T]]/(g_{i}(T))$ and $\displaystyle\prod_{i}g_{i}(T)=g(T)$ for some $g_{i}(T)$’s. Thus we have ${\mathbb{Q}}_{p}[[T]]/g_{i}(T)\cong{\mathbb{Q}}_{p}^{\deg(g_{i})}$ and hence ${\mathcal{H}}_{p}\otimes{\mathbb{Q}}\cong{\mathbb{Q}}_{p}^{\lambda_{p}}$ as linear spaces over ${\mathbb{Q}}_{p}$. The characteristic polynomial of the meridian action (the $t$ action) on ${\mathcal{H}}_{p}\otimes{\mathbb{Q}}$ is $g(t-1)\in{\mathbb{Z}}[t]$, and hence its entropy is given by

[TABLE]

5.2. ${\widehat{{\mathbb{Z}}}}$-covers

Next, we consider branched ${\widehat{{\mathbb{Z}}}}$-covers. Note that the profinite integer ring admits the decomposition ${\widehat{{\mathbb{Z}}}}=\prod_{p}{\mathbb{Z}}_{p}$ by the Chinese remainder theorem. A branched ${\widehat{{\mathbb{Z}}}}$-cover over $(L,S^{3})$ is an inverse system $\{M_{n}\to S^{3}\}_{n}$ of branched ${\mathbb{Z}}/n{\mathbb{Z}}$-covers branched over a link. Let $\tau:H_{1}(S^{3}-L)\to{\widehat{{\mathbb{Z}}}};t_{i}\mapsto z_{i}$, $z_{i}\in{\widehat{{\mathbb{Z}}}}^{\times}$ be a homomorphism. Then the kernels of the composites of $\tau$ and the natural surjections ${\widehat{{\mathbb{Z}}}}\twoheadrightarrow{\mathbb{Z}}/n{\mathbb{Z}}$ define an inverse system of branched ${\mathbb{Z}}/n{\mathbb{Z}}$-covers, and hence a branched ${\widehat{{\mathbb{Z}}}}$-cover associated to $\tau$. We put $A_{L}(t):=(t-1)\Delta_{L}(t^{z_{1}},\ldots,t^{z_{d}})$ and write $A_{L}(t)=(t-1)^{\delta}H_{L}(t)$, $H_{L}(t)\in{\mathbb{Z}}[[t^{{\widehat{{\mathbb{Z}}}}}]]$, $H_{L}(1)\neq 0$. There is a natural isomorphism between $H_{1}(M_{n};{\widehat{{\mathbb{Z}}}})$ and the profinite completion ${\widehat{H}}_{1}(M_{n})$ of $H_{1}(M_{n})$. There is a natural direct sum decomposition ${H}_{1}(M_{n})\overset{\cong}{\to}{\mathbb{Z}}/n{\mathbb{Z}}^{\oplus\delta}\oplus{\widehat{H}}_{1}^{\prime}(M_{n})$ with $\delta\in{\mathbb{N}}$ such that the Fitting ideal of ${\widehat{H}}_{1}^{\prime}(M_{n})$ over ${\widehat{{\mathbb{Z}}}}[t^{{\mathbb{Z}}/n{\mathbb{Z}}}]$ is given by ${\rm Fitt}_{{\mathbb{Z}}[t^{{\mathbb{Z}}/n{\mathbb{Z}}}]}{\widehat{H}}_{1}^{\prime}(M_{n})=(H_{L}(t))\subset{\widehat{{\mathbb{Z}}}}[t^{{\mathbb{Z}}/n{\mathbb{Z}}}].$

We define the completed Alexander module by putting ${\mathcal{H}}:=\varprojlim_{n}H_{1}(M_{n},{\widehat{{\mathbb{Z}}}})$. This module is a finitely generated ${\widehat{{\mathbb{Z}}}}[[t^{{\widehat{{\mathbb{Z}}}}}]]$-module. Indeed, let $z_{i}=(z_{i,n}\mathop{\mathrm{mod}}\nolimits n)_{n}$ with $z_{i,n}\in{\mathbb{Z}}$ for each $i$. Let $h_{\infty,n}$ denote the ${\widehat{{\mathbb{Z}}}}$-cover corresponding to the map $\tau_{n}:H_{1}(S^{3}-L)\mapsto{\widehat{{\mathbb{Z}}}};t_{i}\mapsto z_{i,n}$ for each $n$. Then the completed Alexander module ${\mathcal{H}}_{n}$ of $h_{\infty,n}$ is a finitely generated ${\widehat{{\mathbb{Z}}}}[[t^{{\mathbb{Z}}}]]$-module by a similar argument to [Uek18, Lemma 11]. Since the ranks of ${\mathcal{H}}_{n}$’s are bounded and ${\mathcal{H}}$ is approximated by ${\mathcal{H}}_{n}$’s, ${\mathcal{H}}$ is a finitely generated ${\widehat{{\mathbb{Z}}}}[[t^{{\mathbb{Z}}}]]$-module. We have a natural direct sum decomposition ${\mathcal{H}}\cong{\widehat{{\mathbb{Z}}}}^{\oplus\delta}\oplus{\mathcal{H}}^{\prime}$ such that the Fitting ideal of ${\mathcal{H}}^{\prime}$ is $(H_{L}(t))\subset{\widehat{{\mathbb{Z}}}}[[t^{{\mathbb{Z}}}]]$.

A branched ${\widehat{{\mathbb{Z}}}}$-cover which cannot be obtained from a ${\mathbb{Z}}$-cover may admit $A_{L}(t)$ in ${\mathbb{Z}}[t]$:

Example 5.1**.**

Let $L=K_{1}\cup K_{2}\cup K_{3}\subset S^{3}$ be a 3-component link with $\Delta_{L}(x,y,z)=B(xyz)\in{\mathbb{Z}}[xyz]$. Take $u\in{\widehat{{\mathbb{Z}}}}^{\times}$ with $u\neq\pm 1$. Let $t_{i}$ denote the meridian of $K_{i}$ in $H_{1}(S^{3}-L)$ for each $i$. Consider tha map $\tau:H_{1}(S^{3}-L)\to{\widehat{{\mathbb{Z}}}};t_{1}\mapsto 1,t_{2}\mapsto u,t_{3}\mapsto-u$. Then it defines a branched ${\widehat{{\mathbb{Z}}}}$-cover which cannot be obtained from a ${\mathbb{Z}}$-cover and admits the characteristic polynomial $A_{L}(t)=(1-t)\Delta_{L}(t,t^{u},t^{-u})=(1-t)B(t)\in{\mathbb{Z}}[t]$.

There is only one non-trivial example in the Rolfsen table ([Rol76]): the link $L=8^{3}_{7}$ satisfies $\Delta_{L}(x,y,z)=1-xyz$, $A_{L}(t)=(1-t)\Delta_{L}(t^{u},t^{-u},t)=(1-t)^{2}$.

A ${\widehat{{\mathbb{Z}}}}$-cover is approximated by ${\mathbb{Z}}$-covers, that is, every layer can be obtained as a quotient of a ${\mathbb{Z}}$-cover. Hence a generalization

[TABLE]

of Fox’s formula (Proposition 3.2) holds. Therefore, if $A_{L}(t)\in{\mathbb{Z}}[t]$, then for any $p\leq\infty$, generalizations of asymptotic formulae with use of $p$-adic Mahler measures (Theorems 3.3) hold:

[TABLE]

In addition, if $H_{L}(t):=A_{L}(t)/(t-1)^{d-1}$ has no root on roots of unity, then a generalization of the purely $p$-adic version (Theorem 4.5) for each $p<\infty$ holds:

[TABLE]

In the following, we consider a branched ${\widehat{{\mathbb{Z}}}}$-cover with $A_{L}(t)\in{\mathbb{Z}}[t]$ and study the entropy of the meridian action ($t$-action). Note the decomposition ${\widehat{{\mathbb{Z}}}}[[t^{{\widehat{{\mathbb{Z}}}}}]]\cong\prod_{p}{\mathbb{Z}}_{p}[[t^{{\widehat{{\mathbb{Z}}}}}]]$. Let \operatorname*{\mathchoice{\ooalign{\displaystyle\prod$\crcr$\displaystyle\coprod}}{\ooalign{\textstyle\prod$\crcr$\textstyle\coprod}}{\ooalign{\scriptstyle\prod$\crcr$\scriptstyle\coprod}}{\ooalign{\scriptscriptstyle\prod$\crcr$\scriptscriptstyle\coprod}}}_{p}{\mathbb{Q}}_{p} denotes the restricted product of ${\mathbb{Q}}_{p}$’s with respect to open subgroups ${\mathbb{Z}}_{p}<{\mathbb{Q}}_{p}$.

Here we assume {\mathcal{H}}\otimes{\mathbb{Q}}\cong(\operatorname*{\mathchoice{\ooalign{\displaystyle\prod$\crcr$\displaystyle\coprod}}{\ooalign{\textstyle\prod$\crcr$\textstyle\coprod}}{\ooalign{\scriptstyle\prod$\crcr$\scriptstyle\coprod}}{\ooalign{\scriptscriptstyle\prod$\crcr$\scriptscriptstyle\coprod}}}_{p}{\mathbb{Q}}_{p})^{{\rm deg}(A_{L}(t))}. Then for each prime number $p$, the degree of $A_{L}(t)\mathop{\mathrm{mod}}\nolimits p$ in ${\mathbb{F}}_{p}[t]$ coincides with that of $A_{L}(t)\in{\mathbb{Z}}[t]$. Indeed, since ${\mathcal{H}}$ is a quotient of compact Hausdorff group, it has no $p$-divisible factor. Hence the dimension of ${\mathcal{H}}\otimes{\mathbb{Q}}$ over ${\mathbb{Q}}_{p}$ and that of ${\mathcal{H}}/p{\mathcal{H}}$ over ${\mathbb{F}}_{p}$ must be coincide. Therefore, the leading coefficient of $A_{L}(t)$ is not divisible by any $p$. Since the characteristic polynomial of the $t$-action on each ${\mathbb{Q}}_{p}^{{\rm deg}(A_{L}(t))}$ is $A_{L}(t)$, its $p$-adic entropy is given by $h_{p}=\log\textsc{m}_{p}(A_{L}(t))=0$.

The pontryagin dual of \operatorname*{\mathchoice{\ooalign{\displaystyle\prod$\crcr$\displaystyle\coprod}}{\ooalign{\textstyle\prod$\crcr$\textstyle\coprod}}{\ooalign{\scriptstyle\prod$\crcr$\scriptstyle\coprod}}{\ooalign{\scriptscriptstyle\prod$\crcr$\scriptscriptstyle\coprod}}}_{p}{\mathbb{Q}}_{p} is isomorphic to \operatorname*{\mathchoice{\ooalign{\displaystyle\prod$\crcr$\displaystyle\coprod}}{\ooalign{\textstyle\prod$\crcr$\textstyle\coprod}}{\ooalign{\scriptstyle\prod$\crcr$\scriptstyle\coprod}}{\ooalign{\scriptscriptstyle\prod$\crcr$\scriptscriptstyle\coprod}}}_{p}{\mathbb{Q}}_{p}. We can replace the adèle ring ${\mathbb{A}}_{\mathbb{Q}}$ by \operatorname*{\mathchoice{\ooalign{\displaystyle\prod$\crcr$\displaystyle\coprod}}{\ooalign{\textstyle\prod$\crcr$\textstyle\coprod}}{\ooalign{\scriptstyle\prod$\crcr$\scriptstyle\coprod}}{\ooalign{\scriptscriptstyle\prod$\crcr$\scriptscriptstyle\coprod}}}_{p}{\mathbb{Q}}_{p} in the argument of [LW88, Lemmas 4.3-4.5] with use of results of [Bow71] and [Wal82]. Since the finite set $P$ of prime numbers with contribution is empty, the topological entropy $h$ of the $t$-action on the dual of ${\mathcal{H}}_{\mathbb{Q}}$ is given by $h=0$.

Acknowledgment

I would like to express my sincere gratitude to Mikio Furuta and Yuichiro Taguchi for their precise questions at the thesis defense. I am grateful to Abhijit Champanerkar, Teruhisa Kadokami, Matilde Lalín, Yoshihiko Matsumoto, Yasushi Mizusawa, Takayuki Morisawa, Masanori Morishita, Hirofumi Niibo, Yuji Terashima, and people at the coffee time in Komaba for helpful comments. I would like to thank Tomoki Mihara for precious guidance to $p$-adic analysis. I am also grateful to Dohyeong Kim for inviting me to the beautiful environment of POSTECH IBS-CBP, Noburu Ito, Kouki Taniyama, and Seiken Saito for serving important opportunities to give talks, my family and friends for essential support. I was partially supported by Grant-in-Aid for JSPS Fellows (25-2241).