  Kolyvagin used Heegner points to associate a system of cohomology classes to
an elliptic curve over $\Q$ and conjectured that the system contains a
non-trivial class. His conjecture has profound implications on the structure of
Selmer groups. We provide new computational and theoretical evidence for
Kolyvagin's conjecture. More precisely, we explicitly compute Heegner points
over ring class fields and use these points to verify the conjecture for
specific elliptic curves of rank two. We explain how Kolyvagin's conjecture
implies that if the analytic rank of an elliptic curve is at least two then the
$\Z_p$-corank of the corresponding Selmer group is at least two as well. We
also use explicitly computed Heegner points to produce non-trivial classes in
the Shafarevich-Tate group.
