The rationality problem for four-dimensional linear actions (Q2654726)

Let \(G\) be a finite subgroup of \(\text{GL}_4(\mathbb Q)\). The group \(G\) acts on the rational function field \(\mathbb Q(x_1,x_2,x_3\), \(x_4)\) by \(\mathbb Q\)-automorphisms defined by the linear actions, i.e. if \(\sigma =(a_{ij})_{1 \leq i,j \leq 4} \in G \subset \text{GL}_4(\mathbb Q)\), then \(\sigma (x_j) = \sum_{1 \leq i \leq 4} a_{ij}x_i\). One will like to understand whether the fixed field \(\mathbb Q(x_1,x_2,x_3,x_4)^G\) is \(\mathbb Q\)-rational (i.e. purely transcendental over \(\mathbb Q\)). It was proved by Kitayama that, if \(G\) is a 2-group, then \(\mathbb Q(x_1,x_2,x_3,x_4)^G\) is \(\mathbb Q\)-rational if \(G\) is not isomorphic to the cyclic group of order 8 (Kitayama, preprint); when \(G\) is isomorphic to the cyclic group of order 8, then \(\mathbb Q(x_1,x_2,x_3,x_4)^G\) is not \(\mathbb Q\)-rational by a well known result of Voskresenskii and Lenstra independently [\textit{H. W. Lenstra, jun.}, Invent. Math. 25, 299--325 (1974; Zbl 0292.20010)]. This paper aimed to study the same question for all the finite subgroups of \(\text{GL}_4(\mathbb Q)\). The main result is Theorem 1.1 which states that, for all groups \(G\), \(\mathbb Q(x_1,x_2,x_3,x_4)^G\) is \(\mathbb Q\)-rational except for 6 groups which are labelled as \((4,26,1)\), \((4,33,2)\), \((4,33,3)\), \((4,33,6)\), \((4,33,7)\), \((4,33,11)\); moreover, the answers for \((4,26,1)\), \((4,33,2)\) are negative. But the authors couldn't find the answers for the other four groups. Note that Trepalin of Moscow University obtained similar results (unpublished report). Reviewer's remark: It looks strange that, the proof for 5 cases, i.e. \((4,31, i)\) where \(3 \leq i \leq 7\) (see lines 7-8 from the bottom on page 368), is referred to a preprint of Yamasaki [25], without any other explanation. Up to the writing of this review, the reviewer is still unable to locate this preprint in some published journal or in arXiv. However, the rationality for these groups is just an easy consequence of Lemma 1 and Lemma 5 of the paper by \textit{M. Hajja} and \textit{M.-C. Kang} [8] [J. Algebra 177, No. 2, 511--535 (1995; Zbl 0837.20054)], with the help of Theorem 3.1 in the paper of \textit{Ahmad, M. Hajja} and \textit{M.-C. Kang} [J. Algebra 228, No. 2, 643--658 (2000; Zbl 0993.12003)]. The first paragraph of page 370 gives a proof for the non-rationality of the group \((4,33,2)\) which is a semi-direct product of \(C_3\) with \(C_8\) (with \(C_8\) being the non-normal subgroup). The non-rationality for the group \(C_8\) is due to Voskresenskii and Lenstra as mentioned before. The non-rationality for the group \((4,33,2)\) then follows from Theorem 5.1, Theorem 5.11, and Theorem 3.1 of \textit{D. J. Saltman}'s paper [Adv. Math. 43, 250--283 (1982; Zbl 0484.12004)]; but the authors of this paper referred it to Theorem 5.2.5 of [11] without giving Saltman his due credit.