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

Let k be a field, Tbe an algebraic k-torus, X be a smooth k-compactification of T and Pic (X) over bar be the Picard group of (X) over bar = X x(k) (k) over bar. Hoshi, Kanai and Yamasaki [HKY22] determined H-1 (k, Pic (X) over bar) 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] = n <= 15 and n not equal 12. In this paper, we determine 64 cases with H-1 (k, Pic (X) over bar) not equal 0 and give a necessary and sufficient condition for the Hasse norm principle for K/k with [K: k] = 12. (c) 2022 Elsevier Inc. All rights reserved.