Let *f* be a modular form of even weight on with
associated motive . Let *K* be a
quadratic imaginary field satisfying certain standard conditions. We
improve a result of Nekovár and prove that
if a rational 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
is not divisible by *p*, then the *p*-part of the
Tate-Shafarevic group of over *K*
is trivial.
An important 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.

1991 Mathematics Subject Classification: 11G18, 11F66, 11R34, 14C15.