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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 WC = H that become trivial in all of the completions of K. Thus, in terms of Galois cohomology, it can be written as
This group was introduced by Serge Lang and John Tate and Igor Shafarevich. Cassels introduced the notation Ш, where Ш is the Cyrillic letter "Sha", for Shafarevich, replacing the older notation TS.
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