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Takenori Kataoka

Publications and source records attributed to Takenori Kataoka.

At least 19 recordsLinked to original sources
arXiv2026

An Iwasawa-type asymptotic formula for multiple $\mathbb{Z}_p$-coverings of graphs

For a possibly ramified $\mathbb{Z}_p^d$-covering of connected graphs, we establish an Iwasawa-type asymptotic formula for the growth of the $p$-adic valuations of the complexities. The formula is expressed as a polynomial in $n$ and $p^n$ with explicit leading coefficients $λ$ and $μ$; in particular, we eliminate the error term of the form $O(p^{(d-1)n})$ appearing in earlier work. We then establish a Kida-type formula describing the behavior of $λ$ under a $p$-covering between $\mathbb{Z}_p^d$-coverings, assuming $μ= 0$. Finally, for any fixed $p$ and integer $d \geq 2$, we construct an unramified $\mathbb{Z}_p^d$-covering of a bouquet with prescribed $λ$- and $μ$-invariants.

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arXiv2026

Construction of graph coverings with prescribed Iwasawa invariants

For a $\mathbb{Z}_p$-covering of connected graphs, an analogue of Iwasawa's class number formula describes the growth of the number of spanning trees in terms of Iwasawa $λ$- and $μ$-invariants. In this paper, we show that any pair $(λ, μ)$ can be realized as the Iwasawa invariants of an unramified $\mathbb{Z}_p$-covering of a bouquet, provided that the necessary condition that $λ$ is odd is satisfied. We further show that any pair $(λ, μ)$, without a parity condition, can be realized if we allow ramified $\mathbb{Z}_p$-coverings.

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arXiv2025

On the local equivariant Tamagawa number conjecture for Tate motives

The local equivariant Tamagawa number conjecture (local ETNC) for a motive predicts a precise relationship between the local arithmetic complex and the root numbers which appear in the (conjectural) functional equations of the $L$-functions. In this paper, we prove the local ETNC for the Tate motives under a certain unramified condition at $p$. Our result gives a generalization of the previous works by Burns--Flach and Burns--Sano. Our strategy basically follows those works and builds upon the classical theory of Coleman maps and its generalization by Perrin-Riou.

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arXiv2025

Iwasawa-type asymptotic formula for Taelman class groups of Drinfeld modules

The Taelman class groups associated to Drinfeld modules over function fields serve as an analogue of ideal class groups of number fields. In this paper, we establish an analogue of Iwasawa's asymptotic formula for $\mathbb{Z}_p$-extensions in the context of the Taelman class groups. For this purpose, we naturally introduce and investigate Taelman class groups with moduli.

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arXiv2025

Determining class groups as Galois modules up to equivalence for some nonabelian extensions

In previous papers, the Galois module structure of minus class groups was studied for abelian CM extensions. In this paper, we discuss some nonabelian cases, focusing on metacyclic extensions. For a certain class of these, we obtain a complete description of the Galois modules that occur as minus class groups, modulo a certain equivalence relation on modules, which was introduced earlier by the same authors.

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arXiv2024

Kida's formula for graphs with ramifications

Recently Iwasawa theory for graphs is developing. A significant achievement includes an analogue of Iwasawa class number formula, which describes the asymptotic growth of the numbers of spanning trees for $\mathbb{Z}_p$-coverings of graphs. Moreover, an analogue of Kida's formula concerning the behavior of the $λ$- and $μ$-invariants is obtained for unramified coverings. In this paper, we establish Kida's formula for possibly ramified coverings.

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arXiv2024

Rarity of pseudo-null Iwasawa modules for $p$-adic Lie extensions

In this paper, we obtain a necessary and sufficient condition for the pseudo-nullity of the $p$-ramified Iwasawa module for $p$-adic Lie extension of totally real fields. It is applied to answer the corresponding question for the minus component of the unramified Iwasawa module for CM-fields. The results show that the pseudo-nullity is very rare.

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arXiv2024

On the Galois module structure of minus class groups

The main object of this paper is the minus class groups associated to CM-fields as Galois modules. In a previous article of the authors, we introduced a notion of equivalence for modules and determined the equivalence classes of the minus class groups. In this paper, we show a concrete application of this result. We also study how large a proportion of equivalence classes can be realized as classes of minus class groups.

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arXiv2024

Minimal resolutions of Iwasawa modules

In this paper, we study the module-theoretic structure of classical Iwasawa modules. More precisely, for a finite abelian $p$-extension $K/k$ of totally real fields and the cyclotomic $\mathbb{Z}_p$-extension $K_{\infty}/K$, we consider $X_{K_{\infty},S}={\rm Gal}(M_{K_{\infty},S}/K_{\infty})$ where $S$ is a finite set of places of $k$ containing all ramifying places in $K_{\infty}$ and archimedean places, and $M_{K_{\infty},S}$ is the maximal abelian pro-$p$-extension of $K_{\infty}$ unramified outside $S$. We give lower and upper bounds of the minimal numbers of generators and of relations of $X_{K_{\infty},S}$ as a $\mathbb{Z}_p[[{\rm Gal}(K_{\infty}/k)]]$-module, using the $p$-rank of ${\rm Gal}(K/k)$. This result explains the complexity of $X_{K_{\infty},S}$ as a $\mathbb{Z}_p[[{\rm Gal}(K_{\infty}/k)]]$-module when the $p$-rank of ${\rm Gal}(K/k)$ is large. Moreover, we prove an analogous theorem in the setting that $K/k$ is non-abelian. We also study the Iwasawa adjoint of $X_{K_{\infty},S}$, and the minus part of the unramified Iwasawa module for a CM-extension. In order to prove these theorems, we systematically study the minimal resolutions of $X_{K_{\infty},S}$.

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arXiv2024

Kida's formula via Selmer complexes

In Iwasawa theory, the $λ$, $μ$-invariants of various arithmetic modules are fundamental invariants that measure the size of the modules. Concerning the minus components of the unramified Iwasawa modules, Kida proved a formula that describes the behavior of those invariants with respect to field extensions. Subsequently, many analogues of Kida's formula have been found in various areas in Iwasawa theory. In this paper, we present a novel approach to such analogues of Kida's formula, based on the perspective of Selmer complexes.

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arXiv2023

Fitting ideals of Jacobian groups of graphs

The Jacobian group of a graph is a finite abelian group through which we can study the graph in an algebraic way. When the graph is a finite abelian covering of another graph, the Jacobian group is equipped with the action of the Galois group. In this paper, we study the Fitting ideal of the Jacobian group as a module over the group ring. We also study the corresponding question for infinite coverings. Additionally, this paper includes module-theoretic approach to Iwasawa theory for graphs.

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arXiv2022

Higher codimension Iwasawa theory for elliptic curves with supersingular reduction

Bleher et al. began studying higher codimension Iwasawa theory for classical Iwasawa modules. Subsequently, Lei and Palvannan studied an analogue for elliptic curves with supersingular reduction. In this paper, we obtain a vast generalization of the work of Lei and Palvannan. A key technique is an approach to the work of Bleher et al. that the author previously proposed. For this purpose, we also study the structure of $\pm$-norm subgroups and duality properties of multiply-signed Selmer groups.

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arXiv2022

On Euler systems for motives and Heegner points

We formulate an Iwasawa main conjecture for a higher rank Euler system for a general motive. We prove "one half" of the main conjecture under mild hypotheses. We also formulate a conjecture on "Darmon-type derivatives" of Euler systems and give an application to the Tamagawa number conjecture. Lastly, we specialize our general framework to the setting of Heegner points and give a natural interpretation of the Heegner point main conjecture in terms of rank two Euler systems.

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arXiv2022

Higher codimension behavior in equivariant Iwasawa theory for CM-fields

In classical Iwasawa theory, we mainly study codimension one behavior of arithmetic modules. Relatively recently, F. M. Bleher, T. Chinburg, R. Greenberg, M. Kakde, G. Pappas, R. Sharifi, and M. J. Taylor started studying higher codimension behavior of unramified Iwasawa modules which are conjectured to be pseudo-null. In this paper, by developing a general algebraic theory on perfect complexes, we obtain a new perspective of their work. That allows us to extend the results to equivariant settings and, even in non-equivariant settings, to obtain more refined results concerning the higher codimension behavior.

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arXiv2021

On the minus component of the equivariant Tamagawa number conjecture for $\mathbb{G}_m$

The equivariant Tamagawa number conjecture (hereinafter called the eTNC) predicts close relationships between algebraic and analytic aspects of motives. In this paper, we prove a lot of new cases of the minus component of the eTNC for $\mathbb{G}_m$ and for CM abelian extensions. One of the main results states that the $p$-component of the eTNC is true when there exists at least one $p$-adic prime that is tamely ramified. The fundamental strategy is inspired by the work of Dasgupta and Kakde on the Brumer-Stark conjecture.

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arXiv2021

Fitting ideals of class groups for CM abelian extensions

Let $K/k$ be a finite abelian CM-extension and $T$ a suitable finite set of finite primes of $k$. In this paper, we determine the Fitting ideal of the minus component of the $T$-ray class group of $K$, except for the $2$-component, assuming the validity of the equivariant Tamagawa number conjecture. As an application, we give a necessary and sufficient condition for the Stickelberger element to lie in that Fitting ideal.

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arXiv2020

Fitting ideals in two-variable equivariant Iwasawa theory and an application

We study equivariant Iwasawa theory for two-variable abelian extensions of an imaginary quadratic field. One of the main goals of this paper is to describe the Fitting ideals of Iwasawa modules using $p$-adic $L$-functions. We also provide an application to Selmer groups of elliptic curves with complex multiplication.

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