Finiteness of the Tate–Shafarevich group for elliptic curves over number fields

Prove that the Tate–Shafarevich group Sha(E/K) is finite for every elliptic curve E over a number field K.

Background

The authors’ conditional algorithm for computing rational points on genus 1 curves over number fields assumes the finiteness of the Tate–Shafarevich group of elliptic curves, a classical conjecture in arithmetic geometry. This finiteness would ensure termination of their algorithm that combines Selmer computations with a search for rational points.

They explicitly define the conjecture in the text to fix terminology and assumptions used for their algorithmic consequences.