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 X ‾ be the Picard group of X ‾ = X × k k ‾ where k ‾ is a fixed separable closure of k . Hoshi, Kanai and Yamasaki [HKY22] , [HKY23] determined H 1 ( k , Pic X ‾ ) for norm one tori T = R K / k ( 1 ) ( G m ) 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 = 16 T m ≤ S 16 ( 1 ≤ m ≤ 1954 ) up to conjugacy, we determine 1101 (resp. 774, 31, 37, 1, 1, 9) cases with H 1 ( k , Pic X ‾ ) = 0 (resp. Z / 2 Z , ( Z / 2 Z ) ⊕ 2 , ( Z / 2 Z ) ⊕ 3 , ( Z / 2 Z ) ⊕ 4 , ( Z / 2 Z ) ⊕ 6 , Z / 4 Z ) where G is the Galois group of the Galois closure L / k of K / k . We see that H 1 ( k , Pic X ‾ ) = 0 implies that the Hasse norm principle holds for K / k . In particular, among 22 primitive G = 16 T m cases, i.e. H ≤ G = 16 T m is maximal with [ G : H ] = 16 , we determine exactly 6 cases ( m = 178 , 708 , 1080 , 1329 , 1654 , 1753 ) with H 1 ( k , Pic X ‾ ) ≠ 0 ( ( Z / 2 Z ) ⊕ 2 , Z / 2 Z , ( Z / 2 Z ) ⊕ 2 , Z / 2 Z , Z / 2 Z , Z / 2 Z ). Moreover, we give a necessary and sufficient condition for the Hasse norm principle for K / k with [ K : k ] = 16 for 22 primitive G = 16 T m 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 K / k ( 1 ) ( G m ) over a number field k via Ono's formula where is the Shafarevich-Tate group of T .