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

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