Explicit Heegner points: Kolyvagin's conjecture and non-trivial elements in the Shafarevich-Tate group

Dimitar Jetchev (UC Berkeley), Kristin Lauter (Microsoft), and William Stein (University of Washington).

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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.