http://mathoverflow.net/questions/19760/decomposition-of-tate-shafarevich-groups-in-field-extensions
http://mathoverflow.net/questions/9924/order-of-the-tate-shafarevich-group
arXiv:1108.6310 Counterexamples to the Hasse principle from arXiv Front: math.NT by Wayne Aitken, Franz Lemmermeyer In this article we develop counterexamples to the Hasse principle using only techniques from undergraduate number theory and algebra. By keeping the technical prerequisites to a minimum, we hope to provide a path for nonspecialists to this interesting area of number theory. The counterexamples considered here extend the classical counterexample of Lind and Reichardt. As discussed in an appendix, this type of counterexample is important in the theory of elliptic curves: today they are interpreted as nontrivial elements in the Tate–Shafarevich group.
arXiv:1108.3323 Local-global principles for torsors over arithmetic curves from arXiv Front: math.AG by David Harbater, Julia Hartmann, Daniel Krashen We consider local-global principles for torsors under linear algebraic groups, over function fields of curves over complete discretely valued fields. The obstruction to such a principle is a version of the Tate-Shafarevich group; and we show that it is finite in important cases. Moreover we obtain necessary and sufficient conditions for local-global principles to hold. The proofs use techniques from patching. We also give new applications to quadratic forms and central simple algebras.
nLab page on Tate-Shafarevich group