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Kolyvagin used Heegner points to associate a system of cohomology
classes to an elliptic curve over Q and conjectured that the
system contains a nontrivial 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 apply results of Zhang and others to
deduce that Kolyvagin classes are computable, then explicitly study
Heegner points over ring class fields and Kolyvagin's 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 Zp-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.
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