Locally trivial torsors that are not Weil-Ch\^atelet divisible
  For every prime p we give infinitely many examples of torsors under abelian varieties over Q that are locally trivial but not divisible by p in the Weil-Ch\^atelet group. We also give an example of a locally trivial torsor under an elliptic curve over Q which is not divisible by 4 in the Weil-Ch\^atelet group. This gives a negative answer to a question of Cassels. 
Brendan Creutz
math.NT