Tate-Shafarevich group (reviewed)

The Tate-Shafarevich group of an abelian variety $A$ over a number field $K$ is $$ Ш(A) = \ker\Bigl(H^1(G_K,A) \to \prod_v H^1(G_{K_v},A_{K_v})\Bigr), $$ where $G_K$ is the absolute Galois group of $K$, and $v$ ranges over all places of $K$, including the archimedean places.

It classifies locally solvable principal homogeneous spaces of $A$.

It is a torsion abelian group that is conjectured to be finite.