Degree three unramified cohomology groups and Noether's problem for groups of order $243$

Abstract

Let $k$ be a field and $G$ be a finite group acting on the rational function field $k(x_g : g\in G)$ by $k$-automorphisms defined as $h(x_g)=x_{hg}$ for any $g,h\in G$. We denote the fixed field $k(x_g : g\in G)^G$ by $k(G)$. Noether's problem asks whether $k(G)$ is rational (= purely transcendental) over $k$. It is well-known that if $C(G)$ is stably rational over $C$, then all the unramified cohomology groups $H_[nr}^i(C(G),Q/Z)=0$ for $i \ge 2$. Hoshi, Kang and Kunyavskii [HKK] showed that, for a $p$-group of order $p^5$ ($p$: an odd prime number), $H_[nr}^2(C(G),Q/Z)\neq 0$ if and only if $G$ belongs to the isoclinism family $\Phi_{10}$. When $p$ is an odd prime number, Peyre [Pe3] and Hoshi, Kang and Yamasaki [HKY1] exhibit some $p$-groups $G$ which are of the form of a central extension of certain elementary abelian $p$-group by another one with $H_[nr}^2(C(G),Q/Z)=0$ and $H_[nr}^3(C(G),Q/Z)\neq 0$. However, it is difficult to tell whether $H_[nr}^3(C(G),Q/Z)$ is non-trivial if $G$ is an arbitrary finite group. In this paper, we are able to determine $H_[nr}^3(C(G),Q/Z)$ where $G$ is any group of order $p^5$ with $p=3, 5, 7$. Theorem 1. Let $G$ be a group of order $3^5$. Then $H_[nr}^3(C(G),Q/Z)\neq 0$ if and only if $G$ belongs to $\Phi_7$. Theorem 2. If $G$ is a group of order $3^5$, then the fixed field $C(G)$ is rational if and only if $G$ does not belong to $\Phi_{7}$ and $\Phi_{10}$. Theorem 3. Let $G$ be a group of order $5^5$ or $7^5$. Then $H_[nr}^3(C(G),Q/Z)\neq 0$ if and only if $G$ belongs to $\Phi_6$, $\Phi_7$ or $\Phi_{10}$. Theorem 4. If $G$ is the alternating group $A_n$, the Mathieu group $M_{11}$, $M_{12}$, the Janko group $J_1$ or the group $PSL_2(F_q)$, $SL_2(F_q)$, $PGL_2(F_q)$ (where $q$ is a prime power), then $H_[nr}^d(C(G),Q/Z)=0$ for any $d\ge 2$. Besides the degree three unramified cohomology groups, we compute also the stable cohomology groups.