The rationality problem for finite subgroups of GL_4(Q)

TL;DR

This paper proves that for most finite subgroups of GL_4(Q), the fixed field of their action on rational functions is rational, with only two known exceptions, completing the classification of rationality for all such groups.

Contribution

The paper completes the classification of rationality for fixed fields of all finite subgroups of GL_4(Q), resolving four remaining cases left open by prior work.

Findings

Most fixed fields are rational over Q.

Two specific groups yield non-rational fixed fields.

The complete classification of rationality for these groups is achieved.

Abstract

Let $G$ be a finite subgroup of $GL_4(\bm{Q})$. The group $G$ induces an action on $\bm{Q}(x_1,x_2,x_3,x_4)$, the rational function field of four variables over $\bm{Q}$. Theorem. The fixed subfield $\bm{Q}(x_1,x_2,x_3,x_4)^G:=\{f\in\bm{Q}(x_1,x_2,x_3,x_4):\sigma \cdot f=f$ for any $\sigma\in G\}$ is rational (i.e.\ purely transcendental) over $\bm{Q}$, except for two groups which are images of faithful representations of $C_8$ and $C_3\rtimes C_8$ into $GL_4(\bm{Q})$ (both fixed fields for these two exceptional cases are not rational over $\bm{Q}$). There are precisely 227 such groups in $GL_4(\bm{Q})$ up to conjugation; the answers to the rationality problem for most of them were proved by Kitayama and Yamasaki \cite{KY} except for four cases. We solve these four cases left unsettled by Kitayama and Yamasaki; thus the whole problem is solved completely.