Birational classification of fields of invariants for groups of order $128$

Abstract

Let $G$ be a finite group acting on the rational function field $\mathbb{C}(x_g : g\in G)$ by $\mathbb{C}$-automorphisms $h(x_g)=x_{hg}$ for any $g,h\in G$. Noether's problem asks whether the invariant field $\mathbb{C}(G)=k(x_g : g\in G)^G$ is rational (i.e. purely transcendental) over $\mathbb{C}$. Saltman and Bogomolov, respectively, showed that for any prime $p$ there exist groups $G$ of order $p^9$ and of order $p^6$ such that $\mathbb{C}(G)$ is not rational over $\mathbb{C}$ by showing the non-vanishing of the unramified Brauer group: $Br_{nr}(\mathbb{C}(G))\neq 0$. For $p=2$, Chu, Hu, Kang and Prokhorov proved that if $G$ is a 2-group of order $\leq 32$, then $\mathbb{C}(G)$ is rational over $\mathbb{C}$. Chu, Hu, Kang and Kunyavskii showed that if $G$ is of order 64, then $\mathbb{C}(G)$ is rational over $\mathbb{C}$ except for the groups $G$ belonging to the two isoclinism families $\Phi_{13}$ and $\Phi_{16}$. Bogomolov and B\"ohning's theorem claims that if $G_1$ and $G_2$ belong to the same isoclinism family, then $\mathbb{C}(G_1)$ and $\mathbb{C}(G_2)$ are stably $\mathbb{C}$-isomorphic. We investigate the birational classification of $\mathbb{C}(G)$ for groups $G$ of order 128 with $Br_{nr}(\mathbb{C}(G))\neq 0$. Moravec showed that there exist exactly 220 groups $G$ of order 128 with $Br_{nr}(\mathbb{C}(G))\neq 0$ forming 11 isoclinism families $\Phi_j$. We show that if $G_1$ and $G_2$ belong to $\Phi_{16}, \Phi_{31}, \Phi_{37}, \Phi_{39}, \Phi_{43}, \Phi_{58}, \Phi_{60}$ or $\Phi_{80}$ (resp. $\Phi_{106}$ or $\Phi_{114}$), then $\mathbb{C}(G_1)$ and $\mathbb{C}(G_2)$ are stably $\mathbb{C}$-isomorphic with $Br_{nr}(\mathbb{C}(G_i))\simeq C_2$. Explicit structures of non-rational fields $\mathbb{C}(G)$ are given for each cases including also the case $\Phi_{30}$ with $Br_{nr}(\mathbb{C}(G))\simeq C_2\times C_2$.