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 Pic\,X̄ be the Picard group of X̄=X×ₖk̄ where k̄ is a fixed separable closure of k. Hoshi, Kanai and Yamasaki [HKY22], [HKY23] determined H¹(k, Pic\, X̄) for norm one tori T=R⁽¹⁾K/k(Gₘ) and gave a necessary and sufficient condition for the Hasse norm principle for extensions $K/k$ of number fields with [K:k]≤ 15. In this paper, we treat the case where $[K:k]=16$. Among $1954$ transitive subgroups G=16Tm≤ S₁₆ (1≤ m≤ 1954) up to conjugacy, we determine $1101$ (resp. $774$, $31$, $37$, $1$, $1$, $9$) cases with H¹(k, Pic\, X̄)=0 (resp. $Z/2Z$, (Z/2Z)⊕ 2, (Z/2Z)⊕ 3, (Z/2Z)⊕ 4, (Z/2Z)⊕ 6, $Z/4Z$) where G is the Galois group of the Galois closure $L/k$ of $K/k$. We see that H¹(k, Pic\, X̄)=0 implies that the Hasse norm principle holds for $K/k$. In particular, among $22$ primitive $G=16Tm$ cases, i.e. H≤ G=16Tm is maximal with $[G:H]=16$, we determine exactly $6$ cases $(m=178, 708, 1080, 1329, 1654, 1753)$ with H¹(k, Pic\, X̄)≠ 0 $($$(Z/2Z)⊕ 2$, $Z/2Z$, $(Z/2Z)⊕ 2$, $Z/2Z$, $Z/2Z$, $Z/2Z$). 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 $T=R⁽¹⁾K/k(G_m)$ over a number field $k$ via Ono's formula $τ(T)=1/|Sha(T)|$ where $Sha(T)$ is the Shafarevich-Tate group of $T$.