Selmer groups and the indivisibility of Heegner points

Abstract

For elliptic curves over Q, we prove the p-indivisibility of derived Heegner points for certain prime numbers p, as conjectured by Kolyvagin in 1991. Applications include the refined Birch–Swinerton-Dyer conjecture in the analytic rank one case, and a converse to the theorem of Gross–Zagier and Kolyvagin. A slightly different version of the converse is also proved earlier by Skinner.