arXiv subjects

Jun Ueki

Publications and source records attributed to Jun Ueki.

At least 19 recordsLinked to original sources

arXiv2026

Taut polynomials from finite quotients of fibered hyperbolic 3-manifold groups

We prove that the finite quotients of a fibered hyperbolic 3-manifold group detect the taut polynomials of fibered faces of the Thurston norm balls, whenever the monodromy map is fully-punctured. Toward this, we develop a general framework for the profinite invariance of twisted multivariable Alexander polynomials. We also identify specific hyperbolic one-cusped 3-manifolds that are profinitely rigid, by a strategy using normalized dilatations and the veering census.

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arXiv2026

A Neukirch-Uchida Theorem for 3-Manifolds

The classical Neukirch-Uchida theorem states that the absolute Galois group determines a number field up to isomorphism. We prove an analogue of this theorem for 3-manifolds in the framework of arithmetic topology. We study infinite links in 3-manifolds that behave like the set of primes, satisfying a Chebotarev density property. Relative to such a stably Chebotarev link, we define the absolute Galois group of a 3-manifold as the inverse limit of profinite completions of finite sublink complements. Our main result shows that two branched covers of the three-sphere over a stably Chebotarev link are homeomorphic if and only if their absolute Galois groups are isomorphic via a characteristic-preserving isomorphism. The proof translates the key ideas from the number-theoretic argument into topology, relying on Hilbert ramification theory for infinite covers and local-global principles. In doing so, it also provides a systematic justification for viewing Chebotarev links as the precise topological analogue of prime numbers in anabelian geometry. In addition, we discuss further conditions for links to play the role of prime numbers

math.GT↗

arXiv2026

Positive characteristic analogues of finite algebraic numbers

J.~Rosen introduced the ring $\mathcal{P}^0_{\mathcal{A}}$ of so-called finite algebraic numbers, which may be seen as an analogue of certain periods in the ring $\mathcal{A}=\prod_p \mathbb{Z}/p\mathbb{Z} /\bigoplus_p \mathbb{Z}/p\mathbb{Z}$, $p$ running through all prime numbers. In this article, we introduce its positive characteristic analogue $\mathcal{P}^0_{\mathcal{A}_K}$ over the rational function field $K=\mathbb{F}_q(\theta)$, $q$ being a prime power, and study foundational properties, and provide further scopes.

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arXiv2024

Liminal ${\rm SL}_2\mathbb{Z}_p$-representations and odd-th cyclic covers of genus one two-bridge knots

Let $p$ be a prime number and let $K$ be a genus one two-bridge knot. In the spirit of arithmetic topology, we observe that if $p$ divides the size of the 1st homology group of some odd-th cyclic branched cover of the knot $K$, then its group $\pi_1(S^3-K)$ admits a liminal ${\rm SL}_2\mathbb{Z}_p$-character, where $\mathbb{Z}_p$ denotes the ring of $p$-adic integers. In addition, we discuss the existence of liminal ${\rm SL}_2\mathbb{Z}_p$-representations and give a remark on a general two-bridge knot. In the course of argument, we also point out a constraint for prime numbers dividing certain Lucas-type sequences by using the Legendre symbols.

math.GT↗

arXiv2024

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

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 $\mu$-invariant of a $\mathbb{Z}_p^{\,2}$-cover can be an arbitrary non-negative integer. We also calculate the Iwasawa $\mu$ and $\lambda$-invariants of the Alexander polynomials of all links in Rolfsen's table.

math.GT↗

arXiv2023

Multiplicity of non-acyclic ${\rm SL}_2$-representations and L-functions of the odd-twisted Whitehead links

We study the divisor of the Reidemeister torsion on the variety of irreducible ${\rm SL}_2\mathbb{C}$-characters of certain knots and links, and provide a geometric interpretation of them. We focus in particular on the family of odd-twisted Whitehead links $W_{2n-1}$ and prove that these divisors have multiplicity two. Furthermore, we apply these results to the study of the $L$-functions of the universal deformations of representations over fields with characteristic $p>2$ of these link groups.

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arXiv2022

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

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 $\nu$'s are arbitrarily large.

math.NT↗

arXiv2022

A Hilbert reciprocity law on 3-manifolds

Based on our homological idelic class field theory, we formulate an analogue of the Hilbert reciprocity law on a rational homology 3-sphere endowed with an infinite link, in the spirit of arithmetic topology; We regard the intersection form on the unitary normal bundle of each knot as an analogue of the Hilbert symbol at each prime ideal to formulate the Hilbert reciprocity law, ensuring that cyclic covers of links are analogues of Kummer extensions.

math.GT↗

arXiv2022

Twisted Iwasawa invariants of knots

Let $p$ be a prime number and $m$ an integer coprime to $p$. In the spirit of arithmetic topology, we introduce the notions of the twisted Iwasawa invariants $\lambda, \mu, \nu$ of ${\rm GL}_N$-representations and $\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}_p$-covers of knots. We prove among other things that the set of Iwasawa invariants determine the genus and the fiberedness of a knot, yielding their profinite rigidity. Several intuitive examples are attached. We further prove the $\mu=0$ theorem for ${\rm SL}_2$-representations of twist knot groups and give some remarks.

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arXiv2021

Modular knots, automorphic forms, and the Rademacher symbols for triangle groups

\'{E}.\~Ghys proved that the linking numbers of modular knots and the "missing" trefoil $K_{2,3}$ in $S^3$ coincide with the values of a highly ubiquitous function called the Rademacher symbol for ${\rm SL}_2\mathbb{Z}$. In this paper, we replace ${\rm SL}_2\mathbb{Z}=\Gamma_{2,3}$ by the triangle group $\Gamma_{p,q}$ for any coprime pair $(p,q)$ of integers with $2\leq p<q$. We invoke the theory of harmonic Maass forms for $\Gamma_{p,q}$ to introduce the notion of the Rademacher symbol $\psi_{p,q}$, and provide several characterizations. Among other things, we generalize Ghys's theorem for modular knots around any "missing" torus knot $K_{p,q}$ in $S^3$ and in a lens space.

math.GT↗

arXiv2021

Modular knots obey the Chebotarev law

We study knots which behave like prime numbers. We discuss the planetary link raised from a hyperbolic fibered link in $S^3$ with an emphasis on surgeries, point out certain subtleness, and refine the construction. In addition, we point out a version of McMullen's theorem for the cases over cusped orbifolds and deduce that the family of modular knots around any torus knot $K_{a,b}$ in $S^3$ together with the missing knot $K_{a,b}$ obey the Chebotarev law. Furthermore, we attach several remarks in the view of arithmetic topology.

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arXiv2020

Non-acyclic ${\rm SL}_2$-representations of twist knots, $-3$-Dehn surgeries, and $L$-functions

We study irreducible ${\rm SL}_2$-representations of twist knots. We first determine all non-acyclic ${\rm SL}_2(\mathbb{C})$-representations, which turn out to lie on a line denoted as $x=y$ in $\mathbb{R}^2$. Our main tools are character variety, Reidemeister torsion, and Chebyshev polynomials. We also verify a certain common tangent property, which yields a result on the $L$-functions of universal deformations, that is, the orders of the associated knot modules. Secondly, we prove that a representation is on the line $x=y$ if and only if it factors through the $(-3)$-Dehn surgery, and is non-acyclic if and only if the image of a certain element is of order 3. Finally, we study absolutely irreducible non-acyclic representations $\overline{\rho}$ over a finite field with characteristic $p>2$ to concretely determine all non-trivial $L$-functions $L_{\rho}$ of the universal deformations over a CDVR. We show among other things that $L_{\rho}$ $\dot{=}$ $k_n(x)^2$ holds for a certain series $k_n(x)$ of polynomials.

math.GT↗

arXiv2019

Profinite rigidity for twisted Alexander polynomials

We formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a $\mathbb{Z}$-cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley's parabolic representations of two-bridge knot groups and give a remark on hyperbolic volumes.

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arXiv2019

Chebotarev links are stably generic

We discuss the relationship between two analogues in a 3-manifold of the set of prime ideals in a number field. We prove that if $(K_i)_{i\in \mathbb{N}_{>0}}$ is a sequence of knots obeying the Chebotarev law in the sense of Mazur and McMullen, then $\mathcal{K}=\cup_i K_i$ is a stably generic link in the sense of Mihara. An example we investigate is the planetary link of a fibered hyperbolic finite link in $S^3$. We also observe a Chebotarev phenomenon of knot decomposition in a degree 5 non-Galois subcover of an $A_5$(icosahedral)-cover.

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arXiv2017

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

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.

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arXiv2017

The profinite completions of knot groups determine the Alexander polynomials

We study several properties of the completed group ring $\widehat{\mathbb{Z}}[[t^{\widehat{\mathbb{Z}}}]]$ and the completed Alexander modules of knots. Then we prove that if the profinite completions of the groups of two knots $J$ and $K$ are isomorphic, then their Alexander polynomials $\Delta_J(t)$ and $\Delta_K(t)$ coincide.

math.GT↗

arXiv2016

On the Iwasawa invariants for links and Kida's formula

Analogues of Iwasawa invariants in the context of 3-dimensional topology have been studied by M.~Morishita and others. In this paper, following the dictionary of arithmetic topology, we formulate an analogue of Kida's formula on $\lambda$-invariants in a $p$-extension of $\mathbb{Z}_p$-fields for 3-manifolds. The proof is given in a parallel manner to Iwasawa's second proof, with use of $p$-adic representations of a finite group. In the course of our arguments, we introduce the notion of a branched $\mathbb{Z}_p$-cover as an inverse system of cyclic branched $p$-covers of 3-manifolds, generalize the Iwasawa type formula, and compute the Tate cohomology of 2-cycles explicitly.

math.GT↗

arXiv2015

Id\`elic class field theory for 3-manifolds and very admissible links

We study a topological analogue of id\`elic class field theory for 3-manifolds, in the spirit of arithmetic topology. We firstly introduce the notion of a very admissible link $\mathcal{K}$ in a 3-manifold $M$, which plays a role analogous to the set of primes of a number field. For such a pair $(M,\mathcal{K})$, we introduce the notion of id\`eles and define the id\`ele class group. Then, getting the local class field theory for each knot in $\mathcal{K}$ together, we establish analogues of the global reciprocity law and the existence theorem of id\`elic class field theory.

math.GT↗