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The $p$-adic limits of class numbers in $\mathbb{Z}_p$-towers

Jun Ueki, Hyuga Yoshizaki

TL;DR

The paper establishes that p-adic limits of class numbers in ${f Z}_p$-towers converge both for global fields and for unbranched ${f Z}_p$-covers of 3-manifolds, tying Kisilevsky’s arithmetic results to topological analogues via Iwasawa-type formulas. It then derives an explicit $p$-adic formula for the limit of $p^n$-th cyclic resultants using roots of unity, $p$-adic logarithms, and Iwasawa invariants, and demonstrates how these results specialize to concrete knot families (torus and twist knots) and to elliptic curves over finite fields. The work classifies when the $p$-adic limit lies in ${f Z}$, analyzes how the $ u$-invariant can be made arbitrarily large with small base torsion, and connects these phenomena to Livingston’s results and to function-field analogues via Fox–Weber-type formulas. Overall, it builds a versatile dictionary between arithmetic topology, knot theory, and algebraic curves, providing explicit computations and guiding questions about $p$-adic invariants in towers and covers.

Abstract

This article discusses variants of Weber's class number problem in the spirit of arithmetic topology to connect the results of Sinnott--Kisilevsky and Kionke. Let $p$ be a prime number. We first prove the $p$-adic convergence of class numbers in a $\mathbb{Z}_p$-extension of a global field and a similar result in a $\mathbb{Z}_p$-cover of a compact 3-manifold. Secondly, we establish an explicit formula for the $p$-adic limit of the $p$-power-th cyclic resultants of a polynomial using roots of unity of orders prime to $p$, the $p$-adic logarithm, and the Iwasawa invariants. Finally, we give thorough investigations of torus knots, twist knots, and elliptic curves; we complete the list of the cases with $p$-adic limits being in $\mathbb{Z}$ and find the cases such that the base $p$-class numbers are small and $ν$'s are arbitrarily large.

The $p$-adic limits of class numbers in $\mathbb{Z}_p$-towers

TL;DR

The paper establishes that p-adic limits of class numbers in -towers converge both for global fields and for unbranched -covers of 3-manifolds, tying Kisilevsky’s arithmetic results to topological analogues via Iwasawa-type formulas. It then derives an explicit -adic formula for the limit of -th cyclic resultants using roots of unity, -adic logarithms, and Iwasawa invariants, and demonstrates how these results specialize to concrete knot families (torus and twist knots) and to elliptic curves over finite fields. The work classifies when the -adic limit lies in , analyzes how the -invariant can be made arbitrarily large with small base torsion, and connects these phenomena to Livingston’s results and to function-field analogues via Fox–Weber-type formulas. Overall, it builds a versatile dictionary between arithmetic topology, knot theory, and algebraic curves, providing explicit computations and guiding questions about -adic invariants in towers and covers.

Abstract

This article discusses variants of Weber's class number problem in the spirit of arithmetic topology to connect the results of Sinnott--Kisilevsky and Kionke. Let be a prime number. We first prove the -adic convergence of class numbers in a -extension of a global field and a similar result in a -cover of a compact 3-manifold. Secondly, we establish an explicit formula for the -adic limit of the -power-th cyclic resultants of a polynomial using roots of unity of orders prime to , the -adic logarithm, and the Iwasawa invariants. Finally, we give thorough investigations of torus knots, twist knots, and elliptic curves; we complete the list of the cases with -adic limits being in and find the cases such that the base -class numbers are small and 's are arbitrarily large.
Paper Structure (22 sections, 32 theorems, 40 equations, 2 figures)

This paper contains 22 sections, 32 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Global fields
  3. 3-manifolds
  4. Alternative proofs
  5. Cyclic resultants
  6. Signatures
  7. $p$-adic convergence
  8. Explicit formula
  9. Knots
  10. Alexander polynomial and Fox--Weber's formula
  11. Torus knots
  12. Twist knots
  13. Observations for $K=4_1$
  14. Cases with $\lim |H_1(X_{p^n})_{\rm tor}| \in {\mathbb{Z}}$
  15. Can $\nu$ be large with $r_e$ being small?
  16. ...and 7 more sections

Key Result

Theorem 1

Let $k_{p^\infty}$ be a ${\mathbb{Z}}_{p}$-extension of a global field $k$. Then, the sizes of the class groups ${\rm C}(k_{p^n})$, those of the non-$p$-subgroups ${\rm C}(k_{p^n})_{\text{non-}p}$, and those of the $l$-torsion subgroups ${\rm C}(k_{p^n})_{(l)}$ for each prime number $l$ converge in

Figures (2)

  • Figure : Torus knot $T_{a,b}$
  • Figure : Twist knot $J(2,2m)$

Theorems & Definitions (63)

  • Theorem 1: \ref{['thm.field']}, Kisilevsky1997PJM
  • Theorem 2
  • Theorem 3: \ref{['thm.res.conv']}
  • Theorem 4: \ref{['thm.res']}, a short version
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Remark 2.5
  • Theorem 3.1: $p$-adic convergence
  • ...and 53 more