In arithmetic geometry, the Tate–Shafarevich group Ш of an abelian variety A defined over a number field K consists of the elements of the Weil–Châtelet group W C = H 1 {\displaystyle \mathrm {WC} =H^{1}}, where G K = G a l {\displaystyle G_{K}=\mathrm {Gal} } is the absolute Galois group of K, that become trivial in all of the completions of K. Wikipedia