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 acting on K(x,y,z) by monomial K-automorphisms. We consider the rationality problem of the fixed field KG(x,y,z) under the action of G, namely whether KG(x,y,z) is rational (that is, purely transcendental) over K or not. We may assume that G is a subgroup of GL(3,Z) and the problem is determined up to conjugacy in GL(3,Z). There are 73 conjugacy classes of G in GL(3,Z). By results of Endo–Miyata, Voskresenskiĭ, Lenstra, Saltman, Hajja, Kang and Yamasaki, 8 conjugacy classes of 2-groups in GL(3,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 KG(x,y,z) over K is given. In this paper, we show that the fixed field KG(x,y,z) 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 the 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 KG(x,y,z) is rational over K. We also give an application of the result to 4-dimensional linear Noetherʼs problem.