Tate–Shafarevich group
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In arithmetic geometry, the Tate–Shafarevich group Ш(A/K) of an abelian variety A (or more generally a group scheme) defined over a number field K consists of the elements of the Weil–Châtelet group , where is the absolute Galois group of K, that become trivial in all of the completions of K (i.e., the real and complex completions as well as the p-adic fields obtained from K by completing with respect to all its Archimedean and non Archimedean valuations v). Thus, in terms of Galois cohomology, Ш(A/K) can be defined as
This group was introduced by Serge Lang and John Tate[1] and Igor Shafarevich.[2] Cassels introduced the notation Ш(A/K), where Ш is the Cyrillic letter "Sha", for Shafarevich, replacing the older notation TS or TŠ.