arXiv subjects

Akinari Hoshi

Publications and source records attributed to Akinari Hoshi.

At least 19 recordsLinked to original sources

arXiv2026

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 $τ(T)=p^2$, $p$ or $1$ via Ono's formula $τ(T)=|H^1(k,\widehat{T})|/|Sha(T)|$.

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arXiv2026

Simplest cubic fields with small class number

Let $m\in\mathbb{Z}$ be an integer and $L_m=\mathbb{Q}(α)$ be the simplest cubic field with class number $h_m$ and conductor $\mathfrak{f}_m$ where $α$ 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 $[\mathcal{O}_{L_m}:\mathbb{Z}[α]]=1$ $($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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arXiv2026

Rationality problem for norm one tori of tensor products of étale 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 étale $k$-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 étale $k$-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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arXiv2025

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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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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arXiv2025

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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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 $τ(T)=1$, $1/2$, $1/4$ of $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ over a number field $k$ via Ono's formula $τ(T)=1/|Sha(T)|$ where $Sha(T)$ is the Shafarevich-Tate group of $T$.

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arXiv2024

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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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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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

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

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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