Abstract: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 , Pic X is the Picard group of X = X × k k and ∨ stands for the Pontryagin dual. On the other hand, in 1963, Ono proved that for the norm one torus T = R(1) K/k ( m) of K/k, X(T ) = 0 if and only i… Show more
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