|SATYA MOHIT, Department of Mathematics and Statistics, Queen's University, Kingston, Ontario K7L 3N6, Canada|
|The ABC-Conjecture and bounds for the order of the Tate-Shafarevich group|
We will discuss various equivalent formulations of the ABC-Conjecture in terms of bounds for the order of invariant quantities associated to elliptic curves, particularly, the Tate-Shafarevich group. We will show that, in order to prove the ABC-Conjecture, it suffices to prove the following fact:
For every elliptic curve E, there is some `small' quadratic twist ED whose Tate-Shafarevich group is bounded by .