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Toshiro Hiranouchi

Publications and source records attributed to Toshiro Hiranouchi.

At least 19 recordsLinked to original sources
arXiv2026

Divisibility and torsion in higher Chow groups over arithmetic fields

Let $X$ be a smooth scheme of dimension $d$ over a field $F$. We study the abelian-group structure of the higher Chow groups $CH^{d+i}(X,j)$. For a prime $l$ different from the characteristic of $F$, we prove divisibility and torsion-freeness results when $i\ge$ the $l$-cohomological dimension of $F$. If $X$ is smooth proper and geometrically irreducible, we also study the kernel of the push-forward $CH^{d+i}(X,j)\to CH^i(F,j)$. We apply these results to finite fields, local fields, and global fields.

math.AG↗
arXiv2026

Krasner Completions of Filtered Bands

We introduce the Krasner completion of a band equipped with a decreasing filtration by subgroups of its unit group and give a criterion for reconstruction from its quotient bands. We recover the Krasner--Linzi inverse-limit theorem for valued fields. We also obtain a valuation ring analogue of the Krasner--Linzi theorem and compare Krasner completions of unit bands with adic completion.

math.AC↗
arXiv2026

Galois Symbols for a Jacobian and Multiplicative Groups

Let $C$ be a smooth projective geometrically connected curve over a field $k$ with a $k$-rational point. Let $J$ be the Jacobian variety of $C$. For an integer $r\geq 1$ and a positive integer $n$ prime to the characteristic of $k$, we prove that the Galois symbol map \[ K(k;J,\mathbb{G}_{m},\ldots,\mathbb{G}_{m})/n \to H_{\mathrm{\acute et}}^{r+1}\bigl(k,J[n]\otimes μ_n^{\otimes r}\bigr) \] is injective, where the multiplicative group $\mathbb{G}_{m}$ occurs $r$ times. The proof uses Akhtar's description of higher Chow groups of zero-cycles and the Beilinson--Lichtenbaum theorem. The case $r=1$ recovers a theorem of Spiess.

math.NT↗
arXiv2026

A Hasse principle for the higher Chow groups of curves over a global field

Let $X$ be a smooth projective curve over a global field $F$, and let $V(X)$ denote the kernel of the push-forward map $CH^2(X,1)\to F^\times$. We study the mod-$l$ structure of $V(X)$ by combining Bloch's exact sequence with a Hasse principle in Galois cohomology associated with the mod-$l$ representation of the Jacobian $J$ of $X$. We obtain an exact sequence that describes the kernel and cokernel of the boundary map in terms of local reduction data and the coinvariant quotient $J[l]_{G_F}$. As a consequence, if $\mathrm{End}_{\overline F}(J)=\mathbb{Z}$ and $J$ has semistable reduction of toric dimension one at some place of $F$, then the mod-$l$ boundary map is an isomorphism for all but finitely many primes $l\neq\mathrm{char}(F)$. We also give explicit computations for elliptic curves.

math.NT↗
arXiv2026

Zero-cycles on varieties over a $\mathfrak{B}_s$-field

A field $F$ is a $\mathfrak{B}_s$-field if, for every finite extension $E'/E$ of $F$, the norm map $K_s^M(E')\to K_s^M(E)$ of the Milnor $K$-groups is surjective. In particular, finite fields ($s=1$), local fields, and certain global fields (with $s=2$) satisfy this condition. For such a field $F$ and a $d$-dimensional variety $X$ over $F$, we prove that $CH^{d+n}(X,n)$ is divisible for $n \geq s+1$. Under a suitable condition on the index of $X$, $CH^{d+s}(X,s)$ is isomorphic to the direct sum of the Milnor $K$-group $K_{s}^M(F)$ and a divisible group. As an application, we study the Kato homology groups $KH_0^{(n)}(X,\mathbb{Z}/l^r\mathbb{Z})$ for any prime $l$ different from the characteristic of $F$.

math.NT↗
arXiv2025

Extended differential symbol and the Kato homology groups

Building on our previous work, we investigate an analogue of the differential symbol map used in the Bloch-Gabber-Kato theorem. Within this framework, for an appropriate variety over a field, the higher Chow group corresponds to the 0-th Kato homology group. Inspired by Akhtar's theorem on higher Chow groups, we investigate the structure of the 0-th Kato homology group for varieties over arithmetic fields, including finite fields, local fields, and global fields of positive characteristic. We also express our results in terms of reciprocity sheaves.

math.NT↗
arXiv2024

An additive variant of the differential symbol maps

Our investigation focuses on an additive analogue of the Bloch-Gabber-Kato theorem which establishes a relation between the Milnor $K$-group of a field of positive characteristic and a Galois cohomology group of the field. Extending the Aritin-Schreier-Witt theory, we present an isomorphism from the Mackey product associated with the Witt group and the multiplicative groups to a Galois cohomology group. As a result, we give an expression for the torsion subgroup of the Brauer group of a field, and more generally, the Kato homology groups.

math.KT↗
arXiv2022

Asymptotic behavior of class groups and cyclotomic Iwasawa theory of elliptic curves

In this article, we study a relation between certain quotients of ideal class groups and the cyclotomic Iwasawa module $X_\infty$ of the Pontrjagin dual of the fine Selmer group of an elliptic curve $E$ defined over $\mathbb{Q}$. We consider the Galois extension field $K^E_n$ of $\mathbb{Q}$ generated by coordinates of all $p^n$-torsion points of $E$, and introduce a quotient $A^E_n$ of the $p$-sylow subgroup of the ideal class group of $K^E_n$ cut out by the modulo $p^n$ Galois representation $E[p^n]$. We describe the asymptotic behavior of $A^E_n$ by using the Iwasawa module $X_\infty$. In particular, under certain conditions, we obtain an asymptotic formula as Iwasawa's class number formula on the order of $A^E_n$ by using Iwasawa's invariants of $X_\infty$.

math.NT↗
arXiv2022

Abelian geometric fundamental groups for curves over a $p$-adic field

For a curve $X$ over a $p$-adic field $k$, using the class field theory of $X$ due to S. Bloch and S. Saito we study the abelian geometric fundamental group $π_1^{\mathrm{ab}}(X)^{\mathrm{geo}}$ of $X$. In particular, it is investigated a subgroup of $π_1^{\mathrm{ab}}(X)^{\mathrm{geo}}$ which classifies the geometric and abelian coverings of $X$ which allow possible ramification over the special fiber of the model of $X$. Under the assumptions that $X$ has a $k$-rational point, $X$ has good reduction and its Jacobian variety has good ordinary reduction, we give some upper and lower bounds of this subgroup of $π_1^{\mathrm{ab}}(X)^{\mathrm{geo}}$.

math.NT↗
arXiv2021

Divisibility Results for zero-cycles

Let $X$ be a product of smooth projective curves over a finite unramified extension $k$ of $\mathbb{Q}_p$. Suppose that the Albanese variety of $X$ has good reduction and that $X$ has a $k$-rational point. We propose the following conjecture. The kernel of the Albanese map $CH_0(X)^0\rightarrow\text{Alb}_X(k)$ is $p$-divisible. When $p$ is an odd prime, we prove this conjecture for a large family of products of elliptic curves and certain principal homogeneous spaces of abelian varieties. Using this, we provide some evidence for a local-to-global conjecture for zero-cycles of Colliot-Thélène and Sansuc (\cite{Colliot-Thelene/Sansuc1981}), and Kato and Saito (\cite{Kato/Saito1986}).

math.AG↗
arXiv2019

Galois symbol maps for abelian varieties over a $p$-adic field

We study the Galois symbol map associated to the multiplicative group and an abelian variety which has good ordinary reduction over a $p$-adic field. As a byproduct, one can calculate the "class group" in the view of the class field theory for curves over a $p$-adic field.

math.NT↗
arXiv2018

Class field theory for open curves over local fields, II

As a continuation of arXiv:1412.6888, we study the reciprocity map for an open curve $X$ over a local field of characteristic $p>0$. We determine the $p$-part of the kernel of the reciprocity map after restricting ramification when the rank of $X$ is $0$.

math.NT↗
arXiv2016

A Hermite-Minkowski type theorem of varieties over finite fields

As an application of P. Delgine's theorem (Esnault and Kerz in Acta Math. Vietnam. 37:531-562, 2012) on a finiteness of $l$-adic sheaves on a variety over a finite field, we show the finiteness of étale coverings of such a variety with given degree whose ramification bounded along an effective Cartier divisor. This can be thought of a higher dimensional analogue of the classical Hermite-Minkowski theorem.

math.NT↗
arXiv2016

Class field theory for open curves over local fields

Theory for open curves over a local field. After introducing the reciprocity map, we determine the kernel and the cokernel of this map. In addition to this, the Pontrjagin dual of the reciprocity map is also investigated. This gives the one to one correspondence between the set of finite abelian étale coverings and the set of finite index open subgroups of the idèle class group as in the classical class field theory under some assumptions.

math.NT↗
arXiv2014

An additive variant of Somekawa's $K$-groups and Kähler differentials

We introduce a Milnor type $K$-group associated to commutative algebraic groups over a perfect field. It is an additive variant of Somekawa's $K$-group. We show that the $K$-group associated to the additive group and $q$ multiplicative groups of a field is isomorphic to the space of absolute Kähler differentials of degree $q$ of the field, thus giving us a geometric interpretation of the space of absolute Kähler differentials. We also show that the $K$-group associated to the additive group and Jacobian variety of a curve is isomorphic to the homology group of a certain complex.

math.KT↗