Tate–Shafarevich group

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 ${\displaystyle \mathrm {WC} (A/K)=H^{1}(G_{K},A)}$, where ${\displaystyle G_{K}=\mathrm {Gal} (K^{alg}/K)}$ 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

${\displaystyle \bigcap _{v}\mathrm {ker} \left(H^{1}\left(G_{K},A\right)\rightarrow H^{1}\left(G_{K_{v}},A_{v}\right)\right).}$

This group was introduced by Serge Lang and John Tate and Igor Shafarevich. Cassels introduced the notation Ш(A/K), where Ш is the Cyrillic letter "Sha", for Shafarevich, replacing the older notation TS or TŠ.