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

Abstract

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.