Tate–Shafarevich group

In arithmetic geometry, the Tate–Shafarevich group Ш(A/K), introduced by Lang and Tate (1958) and Shafarevich (1959), 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 WC(A/K) = H¹(G_K, A) that become trivial in all of the completions of K (i.e. the p-adic fields obtained from K, as well as its real and complex completions). Thus, in terms of Galois cohomology, it can be written as

Cassels introduced the notation Ш(A/K), where Ш is the Cyrillic letter "Sha", for Shafarevich, replacing the older notation TS.

Geometrically, the non-trivial elements of the Tate–Shafarevich group can be thought of as the homogeneous spaces of A that have K_v-rational points for every place v of K, but no K-rational point. Thus, the group measures the extent to which the Hasse principle fails to hold for rational equations with coefficients in the field K. Lind (1940) gave an example of such a homogeneous space, by showing that the genus 1 curve ${\displaystyle x^{4}-17=2y^{2}}$ has solutions over the reals and over all p-adic fields, but has no rational points. Selmer (1951) gave many more examples, such as

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