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 $\mathrm{Pic}\bar{X}$ 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 [30], [31] determined $H^1(k,\mathrm{Pic}\bar{X})$ for norm one tori $T=R_{K/k}^{\left(1\right)}\left(G_m\right)$ and gave a necessary and sufficient condition for the Hasse norm principle for extensions $K/k$ 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 $H^1(k,\mathrm{Pic}\bar{X})=0$ (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 $H^1(k,\mathrm{Pic}\bar{X})=0$ implies that the Hasse norm principle holds for $K/k$. 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 $H^1(k,\mathrm{Pic}\bar{X})\ne 0$ (${(Z/2Z)}^{\oplus 2}$, $Z/2Z$, ${(Z/2Z)}^{\oplus 2}$, $Z/2Z$, $Z/2Z$, $Z/2Z$). 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 $\tau \left(T\right)=1$, 1/2, 1/4 of $T=R_{K/k}^{\left(1\right)}\left(G_m\right)$ over a number field k via Ono's formula

where

is the Shafarevich-Tate group of T.