Norm one tori and Hasse norm principle

Let $k$ be a field and $T$ be an algebraic $k$-torus. In 1969, over a global field $k$, Voskresenskii proved that there exists an exact sequence $0\to A(T)\to H^1(k,{\rm Pic}\,\overline{X})^\vee\to Sha(T)\to 0$ where $A(T)$ is the kernel of the weak approximation of $T$, $Sha(T)$ is the Shafarevich-Tate group of $T$, $X$ is a smooth $k$-compactification of $T$, $\overline{X}=X\times_k\overline{k}$, ${\rm Pic}\,\overline{X}$ is the Picard group of $\overline{X}$ and $\vee$ stands for the Pontryagin dual. On the other hand, in 1963, Ono proved that for the norm one torus $T=R^{(1)}_{K/k}(G_m)$ of $K/k$, $Sha(T)=0$ if and only if the Hasse norm principle holds for $K/k$. First, we determine $H^1(k,{\rm Pic}\, \overline{X})$ for algebraic $k$-tori $T$ up to dimension $5$. Second, we determine $H^1(k,{\rm Pic}\, \overline{X})$ for norm one tori $T=R^{(1)}_{K/k}(G_m)$ with $[K:k]=n\leq 15$ and $n\neq 12$. We also show that $H^1(k,{\rm Pic}\, \overline{X})=0$ for $T=R^{(1)}_{K/k}(G_m)$ when the Galois group of the Galois closure of $K/k$ is the Mathieu group $M_n\leq S_n$ with $n=11,12,22,23,24$. Third, we give a necessary and sufficient condition for the Hasse norm principle for $K/k$ with $[K:k]=n\leq 15$ and $n\neq 12$. As applications of the results, we get the group $T(k)/R$ of $R$-equivalence classes over a local field $k$ via Colliot-Thélène and Sansuc's formula and the Tamagawa number $τ(T)$ over a number field $k$ via Ono's formula $τ(T)=|H^1(k,\widehat{T})|/|Sha(T)|$.

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