Rationality problem of three-dimensional monomial group actions

Abstract

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.