arXiv subjects

Akinari Hoshi

Publications and source records attributed to Akinari Hoshi.

At least 37 records · Page 2Linked to original sources

arXiv2019

Degree three unramified cohomology groups and Noether's problem for groups of order $243$

Let $k$ be a field and $G$ be a finite group acting on the rational function field $k(x_g : g\in G)$ by $k$-automorphisms defined as $h(x_g)=x_{hg}$ for any $g,h\in G$. We denote the fixed field $k(x_g : g\in G)^G$ by $k(G)$. Noether's problem asks whether $k(G)$ is rational (= purely transcendental) over $k$. It is well-known that if $C(G)$ is stably rational over $C$, then all the unramified cohomology groups $H_[nr}^i(C(G),Q/Z)=0$ for $i \ge 2$. Hoshi, Kang and Kunyavskii [HKK] showed that, for a $p$-group of order $p^5$ ($p$: an odd prime number), $H_[nr}^2(C(G),Q/Z)\neq 0$ if and only if $G$ belongs to the isoclinism family $Φ_{10}$. When $p$ is an odd prime number, Peyre [Pe3] and Hoshi, Kang and Yamasaki [HKY1] exhibit some $p$-groups $G$ which are of the form of a central extension of certain elementary abelian $p$-group by another one with $H_[nr}^2(C(G),Q/Z)=0$ and $H_[nr}^3(C(G),Q/Z)\neq 0$. However, it is difficult to tell whether $H_[nr}^3(C(G),Q/Z)$ is non-trivial if $G$ is an arbitrary finite group. In this paper, we are able to determine $H_[nr}^3(C(G),Q/Z)$ where $G$ is any group of order $p^5$ with $p=3, 5, 7$. Theorem 1. Let $G$ be a group of order $3^5$. Then $H_[nr}^3(C(G),Q/Z)\neq 0$ if and only if $G$ belongs to $Φ_7$. Theorem 2. If $G$ is a group of order $3^5$, then the fixed field $C(G)$ is rational if and only if $G$ does not belong to $Φ_{7}$ and $Φ_{10}$. Theorem 3. Let $G$ be a group of order $5^5$ or $7^5$. Then $H_[nr}^3(C(G),Q/Z)\neq 0$ if and only if $G$ belongs to $Φ_6$, $Φ_7$ or $Φ_{10}$. Theorem 4. If $G$ is the alternating group $A_n$, the Mathieu group $M_{11}$, $M_{12}$, the Janko group $J_1$ or the group $PSL_2(F_q)$, $SL_2(F_q)$, $PGL_2(F_q)$ (where $q$ is a prime power), then $H_[nr}^d(C(G),Q/Z)=0$ for any $d\ge 2$. Besides the degree three unramified cohomology groups, we compute also the stable cohomology groups.

math.AG↗

arXiv2019

Rationality problem for norm one tori in small dimensions

We classify stably/retract rational norm one tori in dimension $n-1$ for $n=2^e$ $(e\geq 1)$ is a power of $2$ and $n=12, 14, 15$. Retract non-rationality of norm one tori for primitive $G\leq S_{2p}$ where $p$ is a prime number and for the five Mathieu groups $M_n\leq S_n$ $(n=11,12,22,23,24)$ is also given.

math.AG↗

arXiv2019

Rationality problem for norm one tori

We classify stably/retract rational norm one tori in dimension $p-1$ where $p$ is a prime number and in dimension up to ten with some minor exceptions.

math.AG↗

arXiv2018

Multiplicative Invariant Fields of Dimension \le 6

The finite subgroups of $GL_4(\bm{Z})$ are classified up to conjugation in \cite{BBNWZ}; in particular, there exist $710$ non-conjugate finite groups in $GL_4(\bm{Z})$. Each finite group $G$ of $GL_4(\bm{Z})$ acts naturally on $\bm{Z}^{\oplus 4}$; thus we get a faithful $G$-lattice $M$ with ${\rm rank}_\bm{Z} M=4$. In this way, there are exactly $710$ such lattices. Given a $G$-lattice $M$ with ${\rm rank}_\bm{Z} M=4$, the group $G$ acts on the rational function field $\bm{C}(M):=\bm{C}(x_1,x_2,x_3,x_4)$ by multiplicative actions, i.e. purely monomial automorphisms over $\bm{C}$. We are concerned with the rationality problem of the fixed field $\bm{C}(M)^G$. A tool of our investigation is the unramified Brauer group of the field $\bm{C}(M)^G$ over $\bm{C}$. A formula of the unramified Brauer group ${\rm Br}_u(\bm{C}(M)^G)$ for the multiplicative invariant field was found by Saltman in 1990. However, to calculate ${\rm Br}_u(\bm{C}(M)^G)$ for a specific multiplicatively invariant field requires additional efforts, even when the lattice $M$ is of rank equal to $4$. Theorem 1. Among the $710$ finite groups $G$, let $M$ be the associated faithful $G$-lattice with ${\rm rank}_\bm{Z} M=4$, there exist precisely $5$ lattices $M$ with ${\rm Br}_u(\bm{C}(M)^G)\neq 0$. In these situations, $B_0(G)=0$ and thus ${\rm Br}_u(\bm{C}(M)^G)\subset H^2(G,M)$. The {\rm GAP IDs} of the five groups $G$ are {\rm (4,12,4,12), (4,32,1,2), (4,32,3,2), (4,33,3,1), (4,33,6,1)} in {\rm \cite{BBNWZ}} and in {\rm \cite{GAP}}. Theorem 2. There exist $6079$ finite subgroups $G$ in $GL_5(\bm{Z})$. Let $M$ be the lattice with rank $5$ associated to each group $G$. Among these lattices precisely $46$ of them satisfy the condition ${\rm Br}_u(\bm{C}(M)^G)\neq 0$. A similar result for lattices of rank $6$ is found also.

math.AG↗

arXiv2017

Rationality problems for relation modules of dihedral groups

Let D_n be the dihedral group of order 2n where n \ge 2, 1 \to R \to F \to D_n \to 1 be a free presentation of D_n. R^{ab}:=R/[R,R] becomes a \bm{Z}[D_n]-lattice. We will study the module structure and the rationality problem of R^{ab}.

math.NT↗

arXiv2017

Three-dimensional purely quasi-monomial actions

Let $G$ be a finite subgroup of $\mathrm{Aut}_k(K(x_1, \ldots, x_n))$ where $K/k$ is a finite field extension and $K(x_1,\ldots,x_n)$ is the rational function field with $n$ variables over $K$. The action of $G$ on $K(x_1, \ldots, x_n)$ is called quasi-monomial if it satisfies the following three conditions (i) $σ(K)\subset K$ for any $σ\in G$; (ii) $K^G=k$ where $K^G$ is the fixed field under the action of $G$; (iii) for any $σ\in G$ and $1 \leq j \leq n$, $σ(x_j)=c_j(σ)\prod_{i=1}^n x_i^{a_{ij}}$ where $c_j(σ)\in K^\times$ and $[a_{i,j}]_{1\le i,j \le n} \in GL_n(\mathbb{Z})$. A quasi-monomial action is called purely quasi-monomial if $c_j(σ)=1$ for any $σ\in G$, any $1\le j\le n$. When $k=K$, a quasi-monomial action is called monomial. The main problem is that, under what situations, $K(x_1,\ldots,x_n)^G$ is rational (= purely transcendental) over $k$. For $n=1$, the rationality problem was solved by Hoshi, Kang and Kitayama. For $n=2$, the problem was solved by Hajja when the action is monomial, by Voskresenskii when the action is faithful on $K$ and purely quasi-monomial, which is equivalent to the rationality problem of $n$-dimensional algebraic $k$-tori which split over $K$, and by Hoshi, Kang and Kitayama when the action is purely quasi-monomial. For $n=3$, the problem was solved by Hajja, Kang, Hoshi and Rikuna when the action is purely monomial, by Hoshi, Kitayama and Yamasaki when the action is monomial except for one case and by Kunyavskii when the action is faithful on $K$ and purely quasi-monomial. In this paper, we determine the rationality when $n=3$ and the action is purely quasi-monomial except for few cases. As an application, we will show the rationality of some $5$-dimensional purely monomial actions which are decomposable.

math.AG↗

arXiv2017

Rationality problem for algebraic tori

We give the complete stably rational classification of algebraic tori of dimensions $4$ and $5$ over a field $k$. In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank $4$ and $5$ is given. We show that there exist exactly $487$ (resp. $7$, resp. $216$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $4$, and there exist exactly $3051$ (resp. $25$, resp. $3003$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $5$. We make a procedure to compute a flabby resolution of a $G$-lattice effectively by using the computer algebra system GAP. Some algorithms may determine whether the flabby class of a $G$-lattice is invertible (resp. zero) or not. Using the algorithms, we determine all the flabby and coflabby $G$-lattices of rank up to $6$ and verify that they are stably permutation. We also show that the Krull-Schmidt theorem for $G$-lattices holds when the rank $\leq 4$, and fails when the rank is $5$. Indeed, there exist exactly $11$ (resp. $131$) $G$-lattices of rank $5$ (resp. $6$) which are decomposable into two different ranks. Moreover, when the rank is $6$, there exist exactly $18$ $G$-lattices which are decomposable into the same ranks but the direct summands are not isomorphic. We confirm that $H^1(G,F)=0$ for any Bravais group $G$ of dimension $n\leq 6$ where $F$ is the flabby class of the corresponding $G$-lattice of rank $n$. In particular, $H^1(G,F)=0$ for any maximal finite subgroup $G\leq {\rm GL}(n,\mathbb{Z})$ where $n\leq 6$. As an application of the methods developed, some examples of not retract (stably) rational fields over $k$ are given.

math.AG↗

arXiv2015

Degree Three Unramified Cohomology Groups

Let $p$ be an odd prime number. Peyre shows that there is a group $G$ of order $p^{12}$ such that $H_{nr}^3(\bm{C}(G), \bm{Q}/\bm{Z})$ is non-trivial. Using Peyre's method, we are able to prove that the same conclusion is true for some groups of order $p^9$.

math.AG↗

arXiv2014

Rationality problem for quasi-monomial actions

We give a short survey of the rationality problem for quasi-monomial actions which includes Noether's problem and the rationality problem for algebraic tori, and report some results on rationality problem in three recent papers Hoshi, Kang and Kitayama [HKKi], Hoshi, Kang and Kunyavskii [HKKu] and Hoshi and Yamasaki [HY].

math.AG↗

arXiv2014

On Noether's problem for cyclic groups of prime order

Let $k$ be a field and $G$ be a finite group acting on the rational function field $k(x_g\,|\,g\in G)$ by $k$-automorphisms $h(x_g)=x_{hg}$ for any $g,h\in G$. Noether's problem asks whether the invariant field $k(G)=k(x_g\,|\,g\in G)^G$ is rational (i.e. purely transcendental) over $k$. In 1974, Lenstra gave a necessary and sufficient condition to this problem for abelian groups $G$. However, even for the cyclic group $C_p$ of prime order $p$, it is unknown whether there exist infinitely many primes $p$ such that $\mathbb{Q}(C_p)$ is rational over $\mathbb{Q}$. Only known $17$ primes $p$ for which $\mathbb{Q}(C_p)$ is rational over $\mathbb{Q}$ are $p\leq 43$ and $p=61,67,71$. We show that for primes $p< 20000$, $\mathbb{Q}(C_p)$ is not (stably) rational over $\mathbb{Q}$ except for affirmative $17$ primes and undetermined $46$ primes. Under the GRH, the generalized Riemann hypothesis, we also confirm that $\mathbb{Q}(C_p)$ is not (stably) rational over $\mathbb{Q}$ for undetermined $28$ primes $p$ out of $46$.

math.NT↗

arXiv2014

Birational classification of fields of invariants for groups of order $128$

Let $G$ be a finite group acting on the rational function field $\mathbb{C}(x_g : g\in G)$ by $\mathbb{C}$-automorphisms $h(x_g)=x_{hg}$ for any $g,h\in G$. Noether's problem asks whether the invariant field $\mathbb{C}(G)=k(x_g : g\in G)^G$ is rational (i.e. purely transcendental) over $\mathbb{C}$. Saltman and Bogomolov, respectively, showed that for any prime $p$ there exist groups $G$ of order $p^9$ and of order $p^6$ such that $\mathbb{C}(G)$ is not rational over $\mathbb{C}$ by showing the non-vanishing of the unramified Brauer group: $Br_{nr}(\mathbb{C}(G))\neq 0$. For $p=2$, Chu, Hu, Kang and Prokhorov proved that if $G$ is a 2-group of order $\leq 32$, then $\mathbb{C}(G)$ is rational over $\mathbb{C}$. Chu, Hu, Kang and Kunyavskii showed that if $G$ is of order 64, then $\mathbb{C}(G)$ is rational over $\mathbb{C}$ except for the groups $G$ belonging to the two isoclinism families $Φ_{13}$ and $Φ_{16}$. Bogomolov and Böhning's theorem claims that if $G_1$ and $G_2$ belong to the same isoclinism family, then $\mathbb{C}(G_1)$ and $\mathbb{C}(G_2)$ are stably $\mathbb{C}$-isomorphic. We investigate the birational classification of $\mathbb{C}(G)$ for groups $G$ of order 128 with $Br_{nr}(\mathbb{C}(G))\neq 0$. Moravec showed that there exist exactly 220 groups $G$ of order 128 with $Br_{nr}(\mathbb{C}(G))\neq 0$ forming 11 isoclinism families $Φ_j$. We show that if $G_1$ and $G_2$ belong to $Φ_{16}, Φ_{31}, Φ_{37}, Φ_{39}, Φ_{43}, Φ_{58}, Φ_{60}$ or $Φ_{80}$ (resp. $Φ_{106}$ or $Φ_{114}$), then $\mathbb{C}(G_1)$ and $\mathbb{C}(G_2)$ are stably $\mathbb{C}$-isomorphic with $Br_{nr}(\mathbb{C}(G_i))\simeq C_2$. Explicit structures of non-rational fields $\mathbb{C}(G)$ are given for each cases including also the case $Φ_{30}$ with $Br_{nr}(\mathbb{C}(G))\simeq C_2\times C_2$.

math.AG↗

arXiv2014

Noether's problems for groups of order 243

Let $k$ be any field, $G$ be a finite group. Let $G$ act on the rational function field $k(x_g:g\in G)$ by $k$-automorphisms defined by $h\cdot x_g=x_{hg}$ for any $g,h\in G$. Denote by $k(G)=k(x_g:g\in G)^G$ the fixed field. Noether's problem asks, under what situations, the fixed field $k(G)$ will be rational (= purely transcendental) over $k$. According to the data base of GAP there are $10$ isoclinism families for groups of order $243$. It is known that there are precisely $3$ groups $G$ of order $243$ (they consist of the isoclinism family $Φ_{10}$) such that the unramified Brauer group of $\bm{C}(G)$ over $\bm{C}$ is non-trivial. Thus $\bm{C}(G)$ is not rational over $\bm{C}$. We will prove that, if $ζ_9 \in k$, then $k(G)$ is rational over $k$ for groups of order $243$ other than these $3$ groups, except possibly for groups belonging to the isoclinism family $Φ_7$.

math.AG↗

arXiv2014

Class Numbers and Algebraic Tori

Let $p$ be an odd prime number, $D_p$ be the dihedral group of order $2p$, $h_p$ and $h^+_p$ be the class numbers of $\bm{Q}(ζ_p)$ and $\bm{Q}(ζ_p+ ζ_p^{-1})$ respectively. Theorem. $h_p^+=1$ if and only if, for any field $k$ admitting a $D_p$-extension, all the algebraic $D_p$-tori over $k$ are stably rational. A similar result for $h_p=1$ and $C_p$-tori is valid also.

math.NT↗

arXiv2012

Noether's problem and unramified Brauer groups

Let $k$ be any field, $G$ be a finite group acing on the rational function field $k(x_g:g\in G)$ by $h\cdot x_g=x_{hg}$ for any $h,g\in G$. Define $k(G)=k(x_g:g\in G)^G$. Noether's problem asks whether $k(G)$ is rational (= purely transcendental) over $k$. It is known that, if $\bm{C}(G)$ is rational over $\bm{C}$, then $B_0(G)=0$ where $B_0(G)$ is the unramified Brauer group of $\bm{C}(G)$ over $\bm{C}$. Bogomolov showed that, if $G$ is a $p$-group of order $p^5$, then $B_0(G)=0$. This result was disproved by Moravec for $p=3,5,7$ by computer calculations. We will prove the following theorem. Theorem. Let $p$ be any odd prime number, $G$ be a group of order $p^5$. Then $B_0(G)\ne 0$ if and only if $G$ belongs to the isoclinism family $Φ_{10}$ in R. James's classification of groups of order $p^5$.

math.AG↗

arXiv2011

Unramified Brauer groups for groups of order p^5

Let $k$ be any field, $G$ be a finite group acting on the rational function field $k(x_g : g\in G)$ by $h\cdot x_g=x_{hg}$ for any $h,g\in G$. Define $k(G)=k(x_g : g\in G)^G$. Noether's problem asks whether $k(G)$ is rational (= purely transcendental) over $k$. It is known that, if $\bC(G)$ is rational over $\bC$, then $B_0(G)=0$ where $B_0(G)$ is the unramified Brauer group of $\bC(G)$ over $\bC$. Bogomolov showed that, if $G$ is a $p$-group of order $p^5$, then $B_0(G)=0$. This result was disproved by Moravec for $p=3,5,7$ by computer computing. We will give a theoretic proof of the following theorem (i.e. by the traditional bare-hand proof without using computers). Theorem. Let $p$ be any odd prime number. Then there is a group $G$ of order $p^5$ satisfying $B_0(G)\neq 0$ and $G/[G,G] \simeq C_p \times C_p$. In particular, $\bC(G)$ is not rational over $\bC$.

math.AC↗

arXiv2011

On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields

Let $m\geq -1$ be an integer. We give a correspondence between integer solutions to the parametric family of cubic Thue equations \[ X^3-mX^2Y-(m+3)XY^2-Y^3=λ\] where $λ>0$ is a divisor of $m^2+3m+9$ and isomorphism classes of the simplest cubic fields. By the correspondence and R. Okazaki's result, we determine the exactly 66 non-trivial solutions to the Thue equations for positive divisors $λ$ of $m^2+3m+9$. As a consequence, we obtain another proof of Okazaki's theorem which asserts that the simplest cubic fields are non-isomorphic to each other except for $m=-1,0,1,2,3,5,12,54,66,1259,2389$.

math.NT↗

arXiv2011

Rationality problem of three-dimensional monomial group actions

Let $K$ be a field of characteristic not two and $K(x,y,z)$ the rational function field over $K$ with three variables $x,y,z$. Let $G$ be a finite group of acting on $K(x,y,z)$ by monomial $K$-automorphisms. We consider the rationality problem of the fixed field $K(x,y,z)^G$ under the action of $G$, namely whether $K(x,y,z)^G$ is rational (that is, purely transcendental) over $K$ or not. We may assume that $G$ is a subgroup of $\mathrm{GL}(3,\mathbb{Z}) and the problem is determined up to conjugacy in $\mathrm{GL}(3,\mathbb{Z})$. There are 73 conjugacy classes of $G$ in $\mathrm{GL}(3,\mathbb{Z})$. By results of Endo-Miyata, Voskresenski\uı, Lenstra, Saltman, Hajja, Kang and Yamasaki, 8 conjugacy classes of 2-groups in $\mathrm{GL}(3,\mathbb{Z})$ have negative answers to the problem under certain monomial actions over some base field $K$, and the necessary and sufficient condition for the rationality of $K(x,y,z)^G$ over $K$ is given. In this paper, we show that the fixed field $K(x,y,z)^G$ under monomial action of $G$ is rational over $K$ except for possibly negative 8 cases of 2-groups and unknown one case of the alternating group of degree four. Moreover we give explicit transcendental bases of the fixed fields over $K$. For unknown case, we obtain an affirmative solution to the problem under some conditions. In particular, we show that if $K$ is quadratically closed field then $K(x,y,z)^G$ is rational over $K$. We also give an application of the result to 4-dimensional linear Noether's problem.

math.AG↗

arXiv2010

On the simplest sextic fields and related Thue equations

We consider the parametric family of sextic Thue equations \[ x^6-2mx^5y-5(m+3)x^4y^2-20x^3y^3+5mx^2y^4+2(m+3)xy^5+y^6=λ\] where $m\in\mathbb{Z}$ is an integer and $λ$ is a divisor of $27(m^2+3m+9)$. We show that the only solutions to the equations are the trivial ones with $xy(x+y)(x-y)(x+2y)(2x+y)=0$.

math.NT↗