Tate-Shafarevich group (reviewed)

For an abelian variety $A$ over a number field $K$ with absolute Galois group $G_K$, the Tate-Shafarevich group Ш$(A)$ consists of elements of the Weil-Châtelet group $H^1(G_K,A))$ that are trivial in every completion $K_v$ of $K$; here $v$ denotes a place of $K$. In other words $$ Ш(A) = \ker\Bigl(H^1(G_K,A) \to \prod_v H^1(G_{K_v},A_{K_v})\Bigr). $$ It is a torsion abelian group that is conjectured to be finite.