Norm one Tori and Hasse norm principle

Let k k be a field and alttext="upper T"> T encoding="application/x-tex">T an algebraic -torus. In 1969, over global , Voskresenskiǐ proved that there exists exact sequence alttext="0 right-arrow upper A left-parenthesis T right-parenthesis H Superscript 1 Baseline k comma P i c X overbar logical-or Ш 0"> 0 → A stretchy="false">( stretchy="false">) H 1 , Pic ⁡ encoding="application/x-tex">\tau Ono’s StartAbsoluteValue ModifyingAbove With caret EndAbsoluteValue EndAbsoluteValue"> stretchy="false">| ^<!-- ^ (T)=|H^1(k,\widehat {T})|/|\Sha (T)|</mml:annotation>