Appears in collection : A Conference in Arithmetic Algebraic Geometry in Memory of Jan Nekovář

Let E be a rational elliptic curve and p be an odd prime of good ordinary reduction for E. In 1991 Kolyvagin conjectured that the system of cohomology classes derived from Heegner points on the p-adic Tate module of E over an imaginary quadratic field K is non-trivial. I will talk about joint work with A.Burungale, F.Castella, and C.Skinner, where we prove Kolyvagin's conjecture in the cases where an anticyclotomic Iwasawa Main Conjecture for E/K is known. Moreover, our methods also yield a proof of a refinement of Kolyvagin's conjecture expressing the divisibility index of the Heegner point Kolyvagin system in terms of the Tamagawa numbers of E. One of the proof’s key ingredients, on which I will focus during the talk, is a refinement of the Kolyvagin system argument for (anticyclotomic) twists of E studied by Jan Nekovář.