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.
Thus, the group measures the extent to which the Hasse principle fails to hold for rational equations with coefficients in the field K. Carl-Erik Lind gave an example of such a homogeneous space, by showing that the genus 1 curve x4 − 17 = 2y2 has solutions over the reals and over all p-adic fields, but has no rational points.
[8] Theorem 1.1 of Nikolaev [9] treats simple abelian varieties over number fields.
Cassels introduced this for elliptic curves, when A can be identified with  and the pairing is an alternating form.
[12] A choice of polarization on A gives a map from A to Â, which induces a bilinear pairing on Ш(A) with values in Q/Z, but unlike the case of elliptic curves this need not be alternating or even skew symmetric.
For an elliptic curve, Cassels showed that the pairing is alternating, and a consequence is that if the order of Ш is finite then it is a square.
For more general abelian varieties it was sometimes incorrectly believed for many years that the order of Ш is a square whenever it is finite; this mistake originated in a paper by Swinnerton-Dyer,[13] who misquoted one of the results of Tate.
On the other hand building on the results just presented Konstantinou showed that for any squarefree number n there is an abelian variety A defined over Q and an integer m with |Ш| = n ⋅ m2.