If a prime $p$ is outside a finite set of primes depending only on the form $f$, and if the image of the Heegner cycle associated with $K$ in the $p$-adic intermediate jacobian of the motive associated to $f$ is not torsion, then the $p$-part of the Tate-Shafarevich group of this motive is trivial.

The main ingredient of this work is an analysis of the behavior of "Kolyvagin test classes" at primes dividing the level $N$. In addition, certain complications, due to the possibility of $f$ having a Galois conjugate self-twist, have to be dealt with.

Later versions of this work as well as other papers of mine are available here.

Amnon Besser <besser@math.ucla.edu>