One way to exhibit non-trivial elements of the Shafarevich-Tate group of elliptic curves over imaginary quadratic number fields is by
producing locally trivial Kolyvagin classes which are globally non-trivial. These classes are constructed from Heegner points.

We discuss a method for testing whether an explicit element of the Shafarevich-Tate group which is represented by a Kolyvagin class is
globally non-trivial. This is done by determining whether the Cassels pairing of the given class and some other Kolyvagin class is non-zero.
We will describe how to compute the pairing and talk about possible uses in cryptographic applications.