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The Iwasawa invariants of $\mathbb{Z}_p^{\,d}$-covers of links

Sohei Tateno, Jun Ueki

TL;DR

This work extends Cuoco–Monsky’s multivariable Iwasawa framework to ${ ext{Z}}_p^d$-covers of links, establishing asymptotic formulas for $e(H_1)$ in branched and unbranched towers and connecting the growth to invariants $oldsymbol{\\mu}, oldsymbol{\lambda}$ via characteristic elements of Λ-modules. Central technical tools include a multivariable $p$-adic Weierstrass preparation theorem, Monsky’s semi-algebraic analysis, and two fundamental exact sequences that relate the kernel actions to the link exterior’s homology; these yield precise growth formulas and rank/torsion estimates under rank assumptions. The paper further ties these invariants to the Alexander and reduced Alexander polynomials, derives Betti-number polynomial periodicity, and provides explicit examples (notably twisted Whitehead links and Rolfsen-table entries) showing how μ can be prescribed and how the invariants behave in concrete 3-manifolds. Overall, the results deepen the arithmetic-topology analogy by delivering a robust, higher-rank Iwasawa-type theory for links and their covers, with potential connections to Greenberg-type refinements and profinite rigidity.

Abstract

Let $p$ be a prime number and let $d\in \mathbb{Z}_{>0}$. In this paper, following the analogy between knots and primes, we study the $p$-torsion growth in a compatible system of $(\mathbb{Z}/p^n\mathbb{Z})^d$-covers of 3-manifolds and establish several analogues of Cuoco--Monsky's multivariable versions of Iwasawa's class number formula. Our main goal is to establish the Cuoco--Monsky type formula for branched covers of links in rational homology 3-spheres. In addition, we prove the precise formula over integral homology 3-spheres prompted by Greenberg's conjecture. We also derive results on reduced Alexander polynomials and on the Betti number periodicity. Furthermore, we investigate the twisted Whitehead links in $S^3$ and point out that the Iwasawa $μ$-invariant of a $\mathbb{Z}_p^{\,2}$-cover can be an arbitrary non-negative integer. We also calculate the Iwasawa $μ$ and $λ$-invariants of the Alexander polynomials of all links in Rolfsen's table.

The Iwasawa invariants of $\mathbb{Z}_p^{\,d}$-covers of links

TL;DR

This work extends Cuoco–Monsky’s multivariable Iwasawa framework to -covers of links, establishing asymptotic formulas for in branched and unbranched towers and connecting the growth to invariants via characteristic elements of Λ-modules. Central technical tools include a multivariable -adic Weierstrass preparation theorem, Monsky’s semi-algebraic analysis, and two fundamental exact sequences that relate the kernel actions to the link exterior’s homology; these yield precise growth formulas and rank/torsion estimates under rank assumptions. The paper further ties these invariants to the Alexander and reduced Alexander polynomials, derives Betti-number polynomial periodicity, and provides explicit examples (notably twisted Whitehead links and Rolfsen-table entries) showing how μ can be prescribed and how the invariants behave in concrete 3-manifolds. Overall, the results deepen the arithmetic-topology analogy by delivering a robust, higher-rank Iwasawa-type theory for links and their covers, with potential connections to Greenberg-type refinements and profinite rigidity.

Abstract

Let be a prime number and let . In this paper, following the analogy between knots and primes, we study the -torsion growth in a compatible system of -covers of 3-manifolds and establish several analogues of Cuoco--Monsky's multivariable versions of Iwasawa's class number formula. Our main goal is to establish the Cuoco--Monsky type formula for branched covers of links in rational homology 3-spheres. In addition, we prove the precise formula over integral homology 3-spheres prompted by Greenberg's conjecture. We also derive results on reduced Alexander polynomials and on the Betti number periodicity. Furthermore, we investigate the twisted Whitehead links in and point out that the Iwasawa -invariant of a -cover can be an arbitrary non-negative integer. We also calculate the Iwasawa and -invariants of the Alexander polynomials of all links in Rolfsen's table.
Paper Structure (40 sections, 70 theorems, 293 equations)

This paper contains 40 sections, 70 theorems, 293 equations.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. Number theory
  4. Arithmetic topology
  5. Iwasawa-type formula of ${\mathbb{Z}}_{p}$-covers of links
  6. $p$-adic preparations
  7. $p$-adic Weierstrass preparation theorem
  8. Results of Monsky
  9. Characteristic elements
  10. Fitting ideals and Characteristic ideals
  11. Completions
  12. Lemmas on $\Lambda$-modules
  13. Torsion estimates
  14. Rank estimates
  15. Replacing $\mathcal{I}_{p^n}$ by $\mathcal{J}_{p^n}$
  16. ...and 25 more sections

Key Result

Theorem 2.1

Let $k_\infty/k$ be a ${\mathbb{Z}}_{p}$-extension and let $k_{p^n}/k$ denote the ${\mathbb{Z}}/p^n{\mathbb{Z}}$-subextension for each $n\in {\mathbb{Z}}_{\geq 0}$. Then there exist invariants $\mu,\lambda\in {\mathbb{Z}}_{\geq0}$ and $\nu\in{\mathbb{Z}}$, depending only on $k_{\infty}/k$, such that holds.

Theorems & Definitions (144)

  • Theorem 2.1: Iwasawa, Iwasawa1959
  • Theorem 2.2: Cuoco--Monsky, CuocoMonsky1981
  • Theorem 2.3: Monsky Monsky1989
  • Theorem 2.4: Iwasawa-type formula; HillmanMateiMorishita2006, KadokamiMizusawa2008, Ueki2
  • Definition 3.1
  • Example 3.2
  • Example 3.3
  • Lemma 3.4: CuocoMonsky1981
  • Proposition 3.5: CuocoMonsky1981
  • proof
  • ...and 134 more