Genus one curves with a fixed Jacobian can be viewed as elements of the Weil-Chatelet group. We will discuss divisibility questions within this group. This leads us to analyzing the divisibility properties of the Tate-Shafarevich group. There are two related questions:
1. Are the elements of the Tate-Shafarevich group divisible within the Weil-Chatelet group? (Cassels' question.)
2. How does the Tate-Shafarevich group intersect the maximal divisible subgroup of the Weil-Chatelet group? (Bashmakov's question.)
We will discuss our answers to these questions. This is joint work with Jakob Stix (Univ of Frankfurt).