Rationality problem for norm one tori for small non-Galois extensions: Symmetric, alternating and linear group cases including stably rational ${\mathrm{PSL}}_2({\mathbb{F}}_{2^d})$ cases with $d=1,2,3$

Abstract

Let $k$ be a field, $K/k$ be a separable field extension of degree $n$ and $L/k$ be the Galois closure of $K/k$. Let $G=\mathrm{Gal}(L/k)$ and $H=\mathrm{Gal}(L/K)⪇G$ with $[G:H]=n$. Then $H$ is core-free in $G$, i.e. ${\bigcap }_{\sigma \in G}{\sigma }^{-1}H\sigma =\{1\}$. Let $T=R_{K/k}^{(1)}({\mathbb{G}}_m)$ be the norm one torus of $K/k$ with $\dim T=n-1$. We fix $G$ as an abstract group and let $H$ range over all core-free subgroups of $G$ up to conjugacy. For each of the following $15$ groups $G\simeq S_3$, $S_4$, $S_5$, $S_6$, $A_4$, $A_5$, $A_6$, ${\mathrm{GL}}_2({\mathbb{F}}_q)$ $(q=3,4,5)$, ${\mathrm{SL}}_2({\mathbb{F}}_q)$ $(q=3,5,7)$, ${\mathrm{PSL}}_2({\mathbb{F}}_q)$ $(q=7,8)$, by using GAP computations with the aid of PARI/GP, we determine, for any core-free subgroup $H\le G$ up to conjugacy, whether $T$ is stably $k$-rational, not stably but retract $k$-rational, or not retract $k$-rational. The resulting $175$ cases ($15$ are Galois, $160$ are non-Galois) split into $7$, $7$, $161$ cases respectively. We also give alternative proofs for the known cases (i) $H=\{1\}$, i.e. $K/k$ is Galois, and (ii) $[K:k]=[G:H]=n\le 15$ with $G\not\simeq 9T27\simeq {\mathrm{PSL}}_2({\mathbb{F}}_8)$. The $6$ stably $k$-rational cases occur for $G\simeq S_3\simeq {\mathrm{PSL}}_2({\mathbb{F}}_2)$, $A_5\simeq {\mathrm{PSL}}_2({\mathbb{F}}_4)$, ${\mathrm{PSL}}_2({\mathbb{F}}_8)$. Except for the stably $k$-rational case $d=1$ and $H=\{1\}$, for $G\simeq {\mathrm{PSL}}_2({\mathbb{F}}_{2^d})$ with $d=1,2,3$, we prove that $T$ is stably $k$-rational if and only if $T$ is retract $k$-rational if and only if $(C_2{)}^d\le H\le (C_2{)}^d⋊C_{2^d-1}$, and conjecture that this holds for any $d\ge 1$. When $k$ is a global field and $T$ is retract $k$-rational, e.g. $(G,H)\simeq ({\mathrm{PSL}}_2({\mathbb{F}}_7),D_4)$, $({\mathrm{PSL}}_2({\mathbb{F}}_8),(C_2{)}^3)$, $({\mathrm{PSL}}_2({\mathbb{F}}_8),(C_2{)}^3⋊C_7)$, the Hasse norm principle holds for $K/k$.

arXiv comment30 pages. Major revision. Title is modified. Abstract is also modified. Replaced "after some efforts we may get" for ${\mathrm{PSL}}_2({\mathbb{F}}_8)$ case with a reproducible computation