Selmer groups and the indivisibility of Heegner points

For elliptic curves over $\mathbb{Q}$, 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.

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Published 18 December 2014