Local-global questions on curves of genus one

We discuss a few linked questions about curves of genus one, mainly over function fields. We discuss whether a point on an elliptic curve, everywhere locally divisible by $n$, is globally divisible by $n$. We look at whether an element of the Tate-Shafarevich group of an elliptic curve is divisible by $n$ as an element of the Weil-Chatelet group and its relation with finite descent obstructions for curves of genus one. Finally, given a global point on an elliptic curve we look at obstructions for its reduction modulo a place $v$ being a generator over the residue field at $v$, for all $v$.