arXiv subjects

Akinari Hoshi

Publications and source records attributed to Akinari Hoshi.

At least 19 recordsLinked to original sources

arXiv2026

Rationality problem for norm one tori of tensor products of \'etale algebras and Hasse norm principle

Let $k$ be a field. Let $A=\prod_{i=1}^r K_i$ and $B=\prod_{j=1}^s E_j$ be \'etale $k$-algebras where $K_i$ and $E_j$ are finite separable field extensions of $k$ with $[K_i:k]=m_i$ and $[E_j:k]=n_j$. Let $\mathcal{T}_A=R^{(1)}_{A/k}(\mathbb{G}_m)$ be the norm one torus of the \'etale $k$-algebra $A$. We prove that if $\gcd(m_i,n_j\mid 1\leq i\leq r, 1\leq j\leq s)=1$ and $\mathcal{T}_A$ and $\mathcal{T}_B$ are stably $($resp. retract$)$ $k$-rational, then the algebraic $k$-torus $\mathcal{T}_A\otimes \mathcal{T}_B$ and the norm one torus $\mathcal{T}_{A\otimes B}$ are stably $($resp. retract$)$ $k$-rational. In particular, if $k$ is a global field, then the Hasse norm principle holds for $(A\otimes B)/k$. We introduce a new invariant of $G$-lattices, the permutation order, whose triviality is equivalent to invertibility, and use it to study the rationality of tensor products $T_1\otimes T_2$ of algebraic $k$-tori. As an application, we obtain large families of field extensions $K/k$ for which the Hasse norm principle holds.

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arXiv2026

Multiplicative $f$-ic forms on algebraic varieties arising from Thaine's generalized Jacobi sums

We study generalized Jacobi sums, cyclotomic numbers, and $d$-compositions in Thaine's framework, and prove new multiplicative identities extending Davenport and Hasse's lifting theorem from the classical prime-power setting to products of prime powers. As applications, we construct multiplicative forms of degree $f\ge2$, i.e. $f$-ic forms, on complete intersections of $f$-ics. This places Pfister's theory of multiplicative quadratic forms over fields within the broader setting of multiplicative $f$-ic forms on affine algebraic varieties, where new phenomena arise. Moreover, a dense open subset $W \subset V$ carries the structure of an algebraic torus, and the multiplicative form is compatible with the induced group law on $W$.

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arXiv2026

Simplest cubic fields with small class number

Let $m\in\mathbb{Z}$ be an integer and $L_m=\mathbb{Q}(\alpha)$ be the simplest cubic field with class number $h_m$ and conductor $\mathfrak{f}_m$ where $\alpha$ is a root of $f_m(X)=X^3-mX^2-(m+3)X-1$. Let $\mathcal{O}_{L_m}$ be the ring of integers of $L_m$. By using PARI/GP, we determine that if $[\mathcal{O}_{L_m}:\mathbb{Z}[\alpha]]=1$ $($resp. $3$, $27$$)$, i.e. $m^2+3m+9=\mathfrak{f}_m$ $($resp. $3\mathfrak{f}_m$, $27\mathfrak{f}_m$$)$, then there exist exactly $581$ (resp. $80$, $142$) integers $m\geq -1$ such that $h_m\leq 1000$. We also show that if $-1\leq m\leq 10^7$, then $h_m<16$ holds for $138=26+31+11+10+36+21+3$ integers $m$. More precisely, there exist $26$ $($resp. $31$, $11$, $10$, $36$, $21$, $3$$)$ integers $m$ with $-1\leq m\leq 10^7$ such that $h_m=1$ $($resp. $3$, $4$, $7$, $9$, $12$, $13$$)$ which are given explicitly. All computations of the class numbers and class groups of the listed fields are certified unconditionally. Under the GRH $($Generalized Riemann Hypothesis$)$, we confirm that these are the only such integers $m$ in the range $-1\leq m\leq 10^7$. We conjecture that the $26$ integers $m$ obtained above with $h_m=1$ are the only integers $m\geq -1$ with $h_m=1$.

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arXiv2025

Hasse norm principle for Heisenberg extensions of degree $p^3$

Let $k$ be a global field and $p$ be an odd prime number. We give a necessary and sufficient condition for the Hasse norm principle for separable field extensions $K/k$, i.e. the determination of the Shafarevich-Tate group $Sha(T)$ of the norm one tori $T=R^{(1)}_{K/k}(G_m)$ of $K/k$, with $[K:k]=p^3$ or $p^2$ when the Galois group of the Galois closure of $K/k$ is the Heisenberg group $E_p(p^3)\simeq (C_p)^2\rtimes C_p$ of order $p^3$, i.e. the extraspecial group of order $p^3$ with exponent $p$. As a consequence, we get the Tamagawa number $\tau(T)=p^2$, $p$ or $1$ via Ono's formula $\tau(T)=|H^1(k,\widehat{T})|/|Sha(T)|$.

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arXiv2025

Hasse norm principle for metacyclic extensions with trivial Schur multiplier

Let $k$ be a global field, $K/k$ be a finite separable field extension and $L/k$ be the Galois closure of $K/k$ with Galois groups $G={\rm Gal}(L/k)$ and $H={\rm Gal}(L/K)\lneq G$. In 1931, Hasse proved that if $G$ is cyclic, then the Hasse norm principle holds for $K/k$. We show that if $G$ is metacyclic with trivial Schur multiplier $M(G)=0$, then $H$ is cyclic and the Hasse norm principle holds for $K/k$. Some examples of metacyclic, dihedral, quasidihedral, modular, generalized quaternion, extraspecial groups and $Z$-groups $G$ with trivial Schur multiplier $M(G)=0$ are given. These provide new examples which the Hasse norm principle hold for non-Galois extensions $K/k$ whose Galois closure is $L/k$ with metacyclic $G={\rm Gal}(L/k)$ and $M(G)=0$.

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arXiv2024

Norm one tori and Hasse norm principle, III: Degree $16$ case

Let $k$ be a field, $T$ be an algebraic $k$-torus, $X$ be a smooth $k$-compactification of $T$ and ${\rm Pic}\,\overline{X}$ be the Picard group of $\overline{X}=X\times_k\overline{k}$ where $\overline{k}$ is a fixed separable closure of $k$. Hoshi, Kanai and Yamasaki [HKY22], [HKY23] determined $H^1(k,{\rm Pic}\, \overline{X})$ for norm one tori $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ and gave a necessary and sufficient condition for the Hasse norm principle for extensions $K/k$ of number fields with $[K:k]\leq 15$. In this paper, we treat the case where $[K:k]=16$. Among $1954$ transitive subgroups $G=16Tm\leq S_{16}$ $(1\leq m\leq 1954)$ up to conjugacy, we determine $1101$ (resp. $774$, $31$, $37$, $1$, $1$, $9$) cases with $H^1(k,{\rm Pic}\, \overline{X})=0$ (resp. $Z/2Z$, $(Z/2Z)^{\oplus 2}$, $(Z/2Z)^{\oplus 3}$, $(Z/2Z)^{\oplus 4}$, $(Z/2Z)^{\oplus 6}$, $Z/4Z$) where $G$ is the Galois group of the Galois closure $L/k$ of $K/k$. We see that $H^1(k,{\rm Pic}\, \overline{X})=0$ implies that the Hasse norm principle holds for $K/k$. In particular, among $22$ primitive $G=16Tm$ cases, i.e. $H\leq G=16Tm$ is maximal with $[G:H]=16$, we determine exactly $6$ cases $(m=178, 708, 1080, 1329, 1654, 1753)$ with $H^1(k,{\rm Pic}\, \overline{X})\neq 0$ $($$(Z/2Z)^{\oplus 2}$, $Z/2Z$, $(Z/2Z)^{\oplus 2}$, $Z/2Z$, $Z/2Z$, $Z/2Z$). Moreover, we give a necessary and sufficient condition for the Hasse norm principle for $K/k$ with $[K:k]=16$ for $22$ primitive $G=16Tm$ cases. As a consequence of the $22$ primitive $G$ cases, we get the Tamagawa number $\tau(T)=1$, $1/2$, $1/4$ of $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ over a number field $k$ via Ono's formula $\tau(T)=1/|Sha(T)|$ where $Sha(T)$ is the Shafarevich-Tate group of $T$.

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arXiv2023

Rationality problem of two-dimensional quasi-monomial group actions

The rationality problem of two-dimensional purely quasi-monomial actions was solved completely by Hoshi, Kang and Kitayama [HKK]. As a generalization, we solve the rationality problem of two-dimensional quasi-monomial actions under the condition that the actions are defined within the base field. In order to prove the theorem, we give a brief review of the Severi-Brauer variety with some examples and rationality results. We also use a rationality criterion for conic bundles of $\mathbb{P}^1$ over non-closed fields.

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arXiv2023

Rationality problem for norm one tori for dihedral extensions

We give a complete answer to the rationality problem (up to stable $k$-equivalence) for norm one tori $R^{(1)}_{K/k}(\mathbb{G}_m)$ of $K/k$ whose Galois closures $L/k$ are dihedral extensions with the aid of Endo and Miyata [EM75, Theorem 1.5, Theorem 2.3] and Endo [End11, Theorem 2.1]. By using a similar technique, we give refinements of the proof of stably rational cases of Endo and Miyata's theorems as an appendix of the paper.

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arXiv2023

Rationality problem for norm one tori for $A_5$ and ${\rm PSL}_2(\mathbb{F}_8)$ extensions

We give a complete answer to the rationality problem (up to stable $k$-equivalence) for norm one tori $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ of $K/k$ whose Galois closures $L/k$ are $A_5\simeq {\rm PSL}_2(\mathbb{F}_4)$ and ${\rm PSL}_2(\mathbb{F}_8)$ extensions. In particular, we prove that $T$ is stably $k$-rational for $G={\rm Gal}(L/k)\simeq {\rm PSL}_2(\mathbb{F}_{8})$, $H={\rm Gal}(L/K)\simeq (C_2)^3$ and $H\simeq (C_2)^3\rtimes C_7$ where $C_n$ is the cyclic group of order $n$ by using GAP computations with the aid of PARI/GP. Based on the result, we conjecture that $T$ is stably $k$-rational for $G\simeq {\rm PSL}_2(\mathbb{F}_{2^d})$, $(C_2)^d\leq H\leq (C_2)^d\rtimes C_{2^d-1}$. Some other cases $G\simeq A_n$, $S_n$, ${\rm GL}_n(\mathbb{F}_{p^d})$, ${\rm SL}_n(\mathbb{F}_{p^d})$, ${\rm PGL}_n(\mathbb{F}_{p^d})$, ${\rm PSL}_n(\mathbb{F}_{p^d})$ and $H\lneq G$ are also investigated for small $n$ and $p^d$.

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arXiv2022

Norm one tori and Hasse norm principle, II: Degree $12$ case

Let $k$ be a field, $T$ be an algebraic $k$-torus, $X$ be a smooth $k$-compactification of $T$ and ${\rm Pic}\,\overline{X}$ be the Picard group of $\overline{X}=X\times_k\overline{k}$. Hoshi, Kanai and Yamasaki [HKY22] determined $H^1(k,{\rm Pic}\, \overline{X})$ for norm one tori $T=R^{(1)}_{K/k}(G_m)$ and gave a necessary and sufficient condition for the Hasse norm principle for extensions $K/k$ of number fields with $[K:k]=n\leq 15$ and $n\neq 12$. In this paper, we determine $64$ cases with $H^1(k,{\rm Pic}\, \overline{X})\neq 0$ and give a necessary and sufficient condition for the Hasse norm principle for $K/k$ where $[K:k]=12$.

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arXiv2022

Hasse norm principle for $M_{11}$ and $J_1$ extensions

We give a necessary and sufficient condition for the Hasse norm principle for field extensions $K/k$ when the Galois groups ${\rm Gal}(L/k)$ of the Galois closure $L/k$ of $K/k$ are isomorphic to the Mathieu group $M_{11}$ of degree $11$ of order $7920$ or the Janko group $J_1$ of order $175560$ by determining $H^1(k,{\rm Pic}\, \overline{X})=0$ or $\mathbb{Z}/2\mathbb{Z}$ for norm one tori $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ with a smooth $k$-compactification $X$ and $\overline{X}=X\times_k\overline{k}$. The result gives a first step towards understanding the all pictures of the Hasse norm principle for the $26$ sporadic simple groups.

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arXiv2022

Davenport and Hasse's theorems and lifts of multiplication matrices of Gaussian periods

Let $e \geq 2$ be an integer, $p^r$ be a prime power with $p^r \equiv 1\ ({\rm mod}\ e)$ and $η_r(i)$ be Gaussian periods of degree $e$ for ${\mathbb F}_{p^r}$. By the dual form of Davenport and Hasse's lifting theorem on Gauss sums, we establish lifts of the multiplication matrices of the Gaussian periods $η_r(0),\ldots,η_r(e-1)$ which are defined by F. Thaine. We also give some examples of the explicit lifts for prime degree $e$ with $3\leq e\leq 23$ which also illustrate relations among lifts of Jacobi sums, Gaussian periods and multiplication matrices of Gaussian periods.

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arXiv2022

Norm one tori and Hasse norm principle

Let $k$ be a field and $T$ be an algebraic $k$-torus. In 1969, over a global field $k$, Voskresenskii proved that there exists an exact sequence $0\to A(T)\to H^1(k,{\rm Pic}\,\overline{X})^\vee\to Sha(T)\to 0$ where $A(T)$ is the kernel of the weak approximation of $T$, $Sha(T)$ is the Shafarevich-Tate group of $T$, $X$ is a smooth $k$-compactification of $T$, $\overline{X}=X\times_k\overline{k}$, ${\rm Pic}\,\overline{X}$ is the Picard group of $\overline{X}$ and $\vee$ stands for the Pontryagin dual. On the other hand, in 1963, Ono proved that for the norm one torus $T=R^{(1)}_{K/k}(G_m)$ of $K/k$, $Sha(T)=0$ if and only if the Hasse norm principle holds for $K/k$. First, we determine $H^1(k,{\rm Pic}\, \overline{X})$ for algebraic $k$-tori $T$ up to dimension $5$. Second, we determine $H^1(k,{\rm Pic}\, \overline{X})$ for norm one tori $T=R^{(1)}_{K/k}(G_m)$ with $[K:k]=n\leq 15$ and $n\neq 12$. We also show that $H^1(k,{\rm Pic}\, \overline{X})=0$ for $T=R^{(1)}_{K/k}(G_m)$ when the Galois group of the Galois closure of $K/k$ is the Mathieu group $M_n\leq S_n$ with $n=11,12,22,23,24$. Third, we give a necessary and sufficient condition for the Hasse norm principle for $K/k$ with $[K:k]=n\leq 15$ and $n\neq 12$. As applications of the results, we get the group $T(k)/R$ of $R$-equivalence classes over a local field $k$ via Colliot-Thélène and Sansuc's formula and the Tamagawa number $τ(T)$ over a number field $k$ via Ono's formula $τ(T)=|H^1(k,\widehat{T})|/|Sha(T)|$.

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arXiv2021

Birational classification for algebraic tori

We give a stably birational classification for algebraic tori of dimensions $3$ and $4$ over a field $k$. First, we define the weak stably equivalence of algebraic tori and show that there exist $13$ (resp. $128$) weak stably equivalent classes of algebraic tori $T$ of dimension $3$ (resp. $4$) which are not stably rational by computing some cohomological stably birational invariants, e.g. the Brauer-Grothendieck group of $X$ where $X$ is a smooth compactification of $T$, provided by Kunyavskii, Skorobogatov and Tsfasman. We make a procedure to compute such stably birational invariants effectively and the computations are done by using the computer algebra system GAP. Second, we define the $p$-part of the flabby class $[\hat{T}]^{fl}$ as a $\mathbb{Z}_p[{\rm Syl}_p(G)]$-lattice and prove that they are faithful and indecomposable $\mathbb{Z}_p[{\rm Syl}_p(G)]$-lattices unless it vanishes for $p=2$ (resp. $p=2,3$) in dimension $3$ (resp. $4$) via $p$-adic analysis. The $\mathbb{Z}_p$-ranks of them are also given. Third, we give a necessary and sufficient condition for which two not stably rational algebraic tori $T$ and $T^\prime$ of dimensions $3$ (resp. $4$) are stably birationally equivalent in terms of the splitting fields and the weak stably equivalent classes of $T$ and $T^\prime$. In particular, the splitting fields of them should coincide if $\hat{T}$ and $\hat{T}^\prime$ are indecomposable. Forth, for each $7$ cases of not stably but retract rational algebraic tori of dimension $4$, we find an algebraic torus $T^\prime$ of dimension $4$ which satisfies that $T\times_k T^\prime$ is stably rational. Finally, we give a criteria to determine whether two algebraic tori $T$ and $T^\prime$ of general dimensions are stably birationally equivalent when $T$ (resp. $T^\prime$) is stably birationally equivalent to some algebraic torus $T^{\prime\prime}$ of dimension up to $4$.

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arXiv2021

A two-dimensional rationality problem and intersections of two quadrics

Let $k$ be a field with char $k\neq 2$ and $k$ be not algebraically closed. Let $a\in k\setminus k^2$ and $L=k(\sqrt{a})(x,y)$ be a field extension of $k$ where $x,y$ are algebraically independent over $k$. Assume that $σ$ is a $k$-automorphism on $L$ defined by \[ σ: \sqrt{a}\mapsto -\sqrt{a},\ x\mapsto \frac{b}{x},\ y\mapsto \frac{c(x+\frac{b}{x})+d}{y} \] where $b,c,d \in k$, $b\neq 0$ and at least one of $c,d$ is non-zero. Let $L^{\langleσ\rangle}=\{u\in L:σ(u)=u\}$ be the fixed subfield of $L$. We show that $L^{\langleσ\rangle}$ is isomorphic to the function field of a certain surface in $P^4_k$ which is given as the intersection of two quadrics. We give criteria for the $k$-rationality of $L^{\langleσ\rangle}$ by using the Hilbert symbol. As an appendix of the paper, we also give an alternative geometric proof of a part of the result which is provided to the authors by J.-L. Colliot-Thélène.

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arXiv2020

Noether's problem and rationality problem for multiplicative invariant fields: a survey

In this paper, we give a brief survey of recent developments on Noether's problem and rationality problem for multiplicative invariant fields including author's recent papers Hoshi [Hos15] about Noether's problem over Q, Hoshi, Kang and Kunyavskii [HKK13], Chu, Hoshi, Hu and Kang [CHHK15], Hoshi [Hos16] and Hoshi, Kang and Yamasaki [HKY16] about Noether's problem over C, and Hoshi, Kang and Kitayama [HKK14] and Hoshi, Kang and Yamasaki [HKY] about rationality problem for multiplicative invariant fields.

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arXiv2020

On Lecacheux's family of quintic polynomials

Kida, Rikuna and Sato [KRS10] developed a classification theory for Brumer's quintic polynomials via Kummer theory arising from associated elliptic curves. We generalize their results to elliptic curves associated to Lecacheux's quintic $F_{20}$-polynomials instead of Brumer's quintic $D_5$-polynomials.

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arXiv2019

An application of cohomological invariants

Let $G$ be a finite group, $k$ be a field and $G\to GL(V_{\rm reg})$ be the regular representation of $G$ over $k$. Then $G$ acts naturally on the rational function field $k(V_{\rm reg})$ by $k$-automorphisms. Define $k(G)$ to be the fixed field $k(V_{\rm reg})^G$. Noether's problem asks whether $k(G)$ is rational (resp. stably rational) over $k$. When $k=\bQ$ and $G$ contains a normal subgroup $N$ with $G/H\simeq C_8$ (the cyclic group of order $8$), Jack Sonn proves that $\bQ(G)$ is not stably rational over $\bQ$, which is a non-abelian extension of a theorem of Endo-Miyata, Voskresenskii, Lenstra and Saltman for the abelian Noether's problem $\bQ(C_8)$. Using the method of cohomological invariants, we are able to generalize Sonn's theorem as follows. Theorem. Let $G$ be a finite group and $N$ $\lhd$ $G$ such that $G/N\simeq C_{2^n}$ with $n\geq 3$. If $k$ is a field satisfying that ${\rm char}\,k=0$ and $k(ζ_{2^n})/k$ is not a cyclic extension where $ζ_{2^n}$ is a primitive $2^n$-th root of unity, then $k(G)$ is not stably rational (resp. not retract rational) over $k$. \end{abstract}

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