arXiv subjects

Aiichi Yamasaki

Publications and source records attributed to Aiichi Yamasaki.

24 records · Page 2Linked to original sources

arXiv2015

Rationality problem of conic bundles

Let $k$ be a field with char $k \not= 2$, $X$ be an affine surface defined by the equation $z^2=P(x)y^2+Q(x)$ where $P(x), Q(x) \in k[x]$ are separable polynomials. We will investigate the rationality problem of $X$ in terms of the polynomials $P(x)$ and $Q(x)$. The necessary and sufficient condition is $s \leq 3$ with minor exceptions, where $s=s_1+s_2+s_3+s_4$, $s_1$ (resp. $s_2$, resp. $s_3$) being the number of $c \in \overline{k}$ such that $P(c)=0$ and $Q(c) \not\in k(c)^2$ (resp. $Q(c)=0$ and $P(c) \not\in k(c)^2$, resp. $P(c)=Q(c)=0$ and $-\frac{Q}{P}(c) \not\in k(c)^2$). $s_4=0$ or $1$ according to the behavior at $x=\infty$. $X$ is a conic bundle over $\mathbb{P}_k^1$, whose rationality was studied by Iskovskikh. Iskovskikh formulated his results in geometric language. This paper aims to give an algebraic counterpart.

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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 $p^9$.

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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.

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arXiv2013

Rationality problem of generalized Châtelet surfaces

The surface $z^2=ay^2+P(x), \, a \in k, \, P(x) \in k[x]$ is not $k$-rational, if $a \not\in k^2$ and $P(x)$ satisfies some conditions. This result essentially due to Iskovskih but his statement is in terms of algebraic geometry, and not so easy to access for the researchers of the field extension. This paper aims to give a formulation accessible more easily. A necessary and sufficient condition for $k$-rationality is given in terms of $a$ and $P$, assuming that $\ch k \not= 2$ and every irreducible component of $P(x)$ is separable over $k$.

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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.

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arXiv2010

Negative solutions to three-dimensional monomial Noether problem

Three-dimensional monomial Noether problem can have negative solutions for 8 groups by the suitable choice of the coefficients. We find the necessary and sufficient condition for the coefficients to have a negative solution. The results are obtained by two criteria of irrationality using Galois cohomology.

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