Norm one tori and Hasse norm principle, III: Degree $16$ case

Abstract

Let $k$ be a field, $T$ be an algebraic $k$-torus, $X$ be a smooth $k$-compactification of $T$ and ${\rm Pic}\,\overline{X}$ be the Picard group of $\overline{X}=X\times_k\overline{k}$ where $\overline{k}$ is a fixed separable closure of $k$. Hoshi, Kanai and Yamasaki [HKY22], [HKY23] determined $H^1(k,{\rm Pic}\, \overline{X})$ for norm one tori $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ and gave a necessary and sufficient condition for the Hasse norm principle for extensions $K/k$ of number fields with $[K:k]\leq 15$. In this paper, we treat the case where $[K:k]=16$. Among $1954$ transitive subgroups $G=16Tm\leq S_{16}$ $(1\leq m\leq 1954)$ up to conjugacy, we determine $1101$ (resp. $774$, $31$, $37$, $1$, $1$, $9$) cases with $H^1(k,{\rm Pic}\, \overline{X})=0$ (resp. $Z/2Z$, $(Z/2Z)^{\oplus 2}$, $(Z/2Z)^{\oplus 3}$, $(Z/2Z)^{\oplus 4}$, $(Z/2Z)^{\oplus 6}$, $Z/4Z$) where $G$ is the Galois group of the Galois closure $L/k$ of $K/k$. We see that $H^1(k,{\rm Pic}\, \overline{X})=0$ implies that the Hasse norm principle holds for $K/k$. In particular, among $22$ primitive $G=16Tm$ cases, i.e. $H\leq G=16Tm$ is maximal with $[G:H]=16$, we determine exactly $6$ cases $(m=178, 708, 1080, 1329, 1654, 1753)$ with $H^1(k,{\rm Pic}\, \overline{X})\neq 0$ $($$(Z/2Z)^{\oplus 2}$, $Z/2Z$, $(Z/2Z)^{\oplus 2}$, $Z/2Z$, $Z/2Z$, $Z/2Z$). Moreover, we give a necessary and sufficient condition for the Hasse norm principle for $K/k$ with $[K:k]=16$ for $22$ primitive $G=16Tm$ cases. As a consequence of the $22$ primitive $G$ cases, we get the Tamagawa number $τ(T)=1$, $1/2$, $1/4$ of $T=R^{(1)}_{K/k}(\mathbb{G}_m)$ over a number field $k$ via Ono's formula $τ(T)=1/|Sha(T)|$ where $Sha(T)$ is the Shafarevich-Tate group of $T$.