Some cases of four dimensional linear Noether's problem
This paper is a continuation of the paper by \textit{H. Kitayama} and \textit{A. Yamasaki} [J. Math. Kyoto Univ. 49, No. 2, 359--380 (2009; Zbl 1188.13004)]. The reader is referred to the review in Zentralblatt Math for a short account of the rationality problem discussed in Kitayama-Yamasaki's paper and this paper. In the paper of Kitayama and Yamasaki, several cases were left unproved: (i) the groups whose GAP codes are \((4,33,14)\), \((4,33,16)\) and \((4,31,i)\) where \(3\leq i\leq7\), and (ii) the groups \((4,33,3)\), \((4,33,6)\), \((4,33,7)\) and \((4,33,11)\). The authors promised to publish the proof for groups in case (i) in a separate paper; this paper is just the one promised before. The question for groups in case (ii) was solved recently by the reviewer and \textit{J. Zhou} [``The rationality problem for finite subgroups of \(\text{GL}_4(\mathbb Q)\)'', \url{arXiv:1006.1156} (2010)] as shown in the note added in proof of this paper under review. The proof for groups of \((4,31,i)\) where \(3\leq i\leq7\) uses quite a lot of computer computation. For an alternative proof, the reader may find another proof in the paper of the reviewer and J. Zhou mentioned before. The proof for the groups \((4,33,14)\) and \((4,33,16)\) is contained in Section 3. The reader is warned of a mis-numbering in the paper of Kitayama and Yamasaki (p. 378). There the reader was told that the rationality for the group \((4,33,14)\) had been given while the proof for the groups of \((4,33,15)\) and \((4,33,16)\) would be delayed to this paper. This is not true. The fact is what had been solved in Kitayama and Yamasaki's paper was the group \((4,33,15)\) instead of the group \((4,33,14)\). In the Appendix of this paper, the author remarks that the simple group \(G = \text{PSp}(3, 2)\) has a generic polynomial over the rational number field. The proof goes as follows. The group \(G \times C_2\) is isomorphic to the finite reflection group \(E_7\). The ring of invariants associated to the reflection representation for \(E_7\) is a polynomial ring by the Chevalley-Todd-Shepherd Theorem for reflection groups. It follows that the answer to Noether's problem for the group \(E_7\) over the rational number field is affirmative. Thus the group \(E_7\) has a generic polynomial by \textit{D. J. Saltman} [Adv. Math. 43, 250--283 (1982; Zbl 0484.12004); Theorem 5.3]. It follows that the group \(G = \text{PSp}(3, 2)\) has also a generic polynomial by another result of Saltman in the same paper (Theorem 3.1). In this paper the reader is referred for the latter theorem of Saltman to Chapters 1 and 5 of the book by Jensen etc., which seems to be a detour and is inappropriate.
scientific article; zbMATH DE number 5835143