Weil-Ch\^atelet divisible elements in Tate-Shafarevich groups I: The   Bashmakov problem for elliptic curves over Q
  For an abelian variety A over a number field k we discuss the maximal divisibile subgroup of H^1(k,A) and its intersection with the subgroup Sha(A/k). The results are most complete for elliptic curves over Q. 
Mirela \c{C}iperiani and Jakob Stix
math.NT