Norm one Tori and Hasse norm principle

Abstract

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)| .

Norm one tori and Hasse norm principle, II: Degree 12 case

“…Kunyavskii [Kun90] gave a rational (stably rational, retract rational) classification of 3-dimensional algebraic k-tori. Hoshi and Yamasaki [HY17, Theorem 1.9 and Theorem 1.12] classified stably/retract rational algebraic k-tori of dimension 4 and 5 (see also [HKY22, Section 1]).…”

“…Gurak [Gur78a] showed that the Hasse norm principle holds for Galois extension K/k if all the Sylow subgroups of Gal(K/k) are cyclic. Note that this also follows from Theorem 3.6 and the retract k-rationality of T = R For non-Galois extension K/k of degree n, the Hasse norm principle was investigated by Bartels [Bar81a] (holds for n = p; prime), [Bar81b] (holds for G ≃ D n ), Voskresenskii and Kunyavskii [VK84] (holds for G ≃ S n ), Kunyavskii [Kun84] (n = 4), Drakokhrust and Platonov [DP87] (n = 6), Macedo [Mac20] (holds for G ≃ A n (n = 4)), Macedo and Newton [MN22] (G ≃ A 4 , S 4 , A 5 , S 5 , A 6 , A 7 (general n)), Hoshi, Kanai and Yamasaki [HKY22] (n ≤ 15 (n = 12)) (holds for G ≃ M n (n = 11, 12, 22, 23, 24; 5 Mathieu groups)), [HKY23] (n = 12), [HKY] (G ≃ M 11 , J 1 (general n)), Hoshi and Yamasaki [HY2] (holds for G ≃ PSL 2 ( 7 ) (n = 21), PSL 2 ( 8 ) (n = 63)) where G = Gal(L/k) and L/k is the Galois closure of K/k. Recall that the case where n = p also follows from Theorem 3.6 and the retract k-rationality of T = R Let T = R…”

Hasse norm principle for $M_{11}$ and $J_1$ extensions

“…The Hasse norm principle for Galois extensions K/k was investigated by Gerth [Ger77], [Ger78] and Gurak [Gur78a], [Gur78b], [Gur80] (see also [PR94,). For non-Galois extension K/k, the Hasse norm principle was investigated by Bartels [Bar81a] Macedo [Mac20] (Gal(L/k) ≃ A n ) where L/k be the Galois closure of K/k, Macedo and Newton [MN22], Hoshi, Kanai and Yamasaki [HKY22], [HKY] (see also [HKY22…”

“…For the flabby class [J G/H ] f l of the Chevalley module J G/H = (I G/H ) • = Hom (I G/H , ) ≃ T = Hom(T, m ) where I G/H = Ker ε and ε : [G/H] → is the argumentation map, as in Table 1 and Table 2, see the related previous papers [HY17], [HHY20], [HY21], [HKY22], [HKY]. It turns out that there exist 38 = 25 + 13 subgroups H G up to conjugacy and 25 (resp.…”

“…In Section 2, we recall Drakokhrust and Platonov's method for the Hasse norm principle for K/k. In Section 3, we give the proof of Theorem 1.6 and Theorem 1.7 using Drakokhrust and Platonov's method with the aid of GAP computations developed by Hoshi, Kanai and Yamasaki [HKY22], [HKY]. In Section 4, GAP computations which are used in the proof of Theorem 1.6 and Theorem 1.7 are given.…”