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 → A(T ) → H 1 (k, Pic X) ∨ → X(T ) → 0 where A(T ) is the kernel of the weak approximation of T , X(T ) is the Shafarevich-Tate group of T , X is a smooth k-compactification of T , X = X × k k, Pic X is the Picard group of X and ∨ stands for the Pontryagin dual. In 1984, Kunyavskii showed that, among 73 cases of 3-dimensional k-tori T , there exist exactly 2 cases satisfy H 1 (k, Pic X) = 0. The two tori T satisfying H 1 (k, Pic X) ≃ /2 are of special type called norm one tori. On the other hand, in 1963, Ono proved that X(T ) = 0 if and only if the Hasse norm principle holds for K/k where T = R(1) K/k ( m) is the norm one torus of K/k. First, we show that, among 710 cases of 4-dimensional algebraic k-tori T , there exist exactly 2 (resp. 20, 688) cases with). Among 6079 cases of 5dimensional algebraic k-tori T , there exist exactly 11 (resp. 263, 5805) cases withwith [K : k] = n ≤ 15 and n = 12. We also show that H 1 (k, Pic X) = 0 for T = R (1) K/k ( m) when the Galois group of the Galois closure of K/k is the Mathieu group Mn ≤ Sn 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 ≤ 15 and n = 12. As applications of the results, we get the group T (k)/R of R-equivalence classes over a local field k via the formula T (k)/R ≃ H 1 (G, S) of Colliot-Thélène and Sansuc where 1 → S → Q → T → 1 is a flabby resolution of T and S is the character module of a torus S, and the Tamagawa number τ (T ) over a number field k via Ono's formula τ (T ) = |H 1 (k, T )|/|X(T )|.
Let k k be a field and T T be an algebraic k k -torus. In 1969, over a global field k k , Voskresenskiǐ proved that there exists an exact sequence 0 → A ( T ) → H 1 ( k , Pic X ¯ ) ∨ → Ш ( T ) → 0 0\to A(T)\to H^1(k,\operatorname {Pic}\overline {X})^\vee \to \Sha (T)\to 0 where A ( T ) A(T) is the kernel of the weak approximation of T T , Ш ( T ) \Sha (T) is the Shafarevich-Tate group of T T , X X is a smooth k k -compactification of T T , X ¯ = X × k k ¯ \overline {X}=X\times _k\overline {k} , Pic X ¯ \operatorname {Pic}\overline {X} is the Picard group of X ¯ \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 K / k ( 1 ) ( G m ) T=R^{(1)}_{K/k}(\mathbb {G}_m) of K / k K/k , Ш ( T ) = 0 \Sha (T)=0 if and only if the Hasse norm principle holds for K / k K/k . First, we determine H 1 ( k , Pic X ¯ ) H^1(k,\operatorname {Pic} \overline {X}) for algebraic k k -tori T T up to dimension 5 5 . Second, we determine H 1 ( k , Pic X ¯ ) H^1(k,\operatorname {Pic} \overline {X}) for norm one tori T = R K / k ( 1 ) ( G m ) T=R^{(1)}_{K/k}(\mathbb {G}_m) with [ K : k ] = n ≤ 15 [K:k]=n\leq 15 and n ≠ 12 n\neq 12 . We also show that H 1 ( k , Pic X ¯ ) = 0 H^1(k,\operatorname {Pic} \overline {X})=0 for T = R K / k ( 1 ) ( G m ) T=R^{(1)}_{K/k}(\mathbb {G}_m) when the Galois group of the Galois closure of K / k K/k is the Mathieu group M n ≤ S n M_n\leq S_n with n = 11 , 12 , 22 , 23 , 24 n=11,12,22,23,24 . Third, we give a necessary and sufficient condition for the Hasse norm principle for K / k K/k with [ K : k ] = n ≤ 15 [K:k]=n\leq 15 and n ≠ 12 n\neq 12 . As applications of the results, we get the group T ( k ) / R T(k)/R of R R -equivalence classes over a local field k k via Colliot-Thélène and Sansuc’s formula and the Tamagawa number τ ( T ) \tau (T) over a number field k k via Ono’s formula τ ( T ) = | H 1 ( k , T ^ ) | / | Ш ( T ) | \tau (T)=|H^1(k,\widehat {T})|/|\Sha (T)| .
Let k be a field, T be an algebraic k-torus, X is a smooth k-compactification of T and Pic X is the Picard group of X = X × k k. Hoshi, Kanai and Yamasaki [HKY] determined H 1 (k, Pic X) for norm one tori T = R(1) K/k ( m) and gave a necessary and sufficient condition for the Hasse norm principle for extensions K/k of number fields with [K : k] = n ≤ 15 and n = 12. In this paper, we determine 64 cases with H 1 (k, Pic X) = 0 and give a necessary and sufficient condition for the Hasse norm principle for K/k where [K : k] = 12.(1) K/k ( m ) be the norm one torus of K/k of dimension 11 and X be a smooth k-compactification of T . Then H 1 (k, Pic X) = 0 if and only if G is given as in Table 1. In particular, if k is a number field and L/k is an unramified extension, then A(T ) = 0 and H 1 (k, Pic X) ≃ X(T ).In Table 1, V 4 ≃ C 2 × C 2 is the Klein four group, Q 8 is the quaternion group of order 8, PSL 2 ( 11 ) is the projective special linear group of degree 2 over the finite field 11 of 11 elements, and S n (m) and A n (m) mean that S n (m) ≃ S n = mT x ≤ S m and A n (m) ≃ A n = mT x ≤ S m respectively.
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