Rationality problem for norm one tori for $A_5$ and ${\rm PSL}_2(\mathbb{F}_8)$ extensions

TL;DR

This paper investigates the rationality problem for norm one tori associated with specific Galois extensions, providing complete solutions for certain cases involving $A_5$ and ${ m PSL}_2(_8)$, and conjecturing broader patterns.

Contribution

It offers a complete analysis of the rationality problem for norm one tori in specific Galois extension cases, including computational proofs and conjectures for general groups.

Findings

Proves stable $k$-rationality for certain ${ m PSL}_2(_{8})$ cases

Uses computational tools like GAP and PARI/GP for proofs

Conjectures broader stable $k$-rationality patterns for groups ${ m PSL}_2(_{2^d})$

Abstract

We give a complete answer to the rationality problem (up to stable $k$-equivalence) for norm one tori $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ of $K/k$ whose Galois closures $L/k$ are $A_5\simeq {\rm PSL}_2(\mathbb{F}_4)$ and ${\rm PSL}_2(\mathbb{F}_8)$ extensions. In particular, we prove that $T$ is stably $k$-rational for $G={\rm Gal}(L/k)\simeq {\rm PSL}_2(\mathbb{F}_{8})$, $H={\rm Gal}(L/K)\simeq (C_2)^3$ and $H\simeq (C_2)^3\rtimes C_7$ where $C_n$ is the cyclic group of order $n$ by using GAP computations with the aid of PARI/GP. Based on the result, we conjecture that $T$ is stably $k$-rational for $G\simeq {\rm PSL}_2(\mathbb{F}_{2^d})$, $(C_2)^d\leq H\leq (C_2)^d\rtimes C_{2^d-1}$. Some other cases $G\simeq A_n$, $S_n$, ${\rm GL}_n(\mathbb{F}_{p^d})$, ${\rm SL}_n(\mathbb{F}_{p^d})$, ${\rm PGL}_n(\mathbb{F}_{p^d})$, ${\rm PSL}_n(\mathbb{F}_{p^d})$ and $H\lneq G$ are also investigated for small $n$ and $p^d$.