Exceptional zeros for Heegner points and p-converse to the theorem of Gross-Zagier and Kolyvagin

We prove a p-converse to the theorem of Gross-Zagier and Kolyvagin for elliptic curves E/Q at primes p>3 of multiplicative reduction. Two key ingredients in the argument are an extension to this setting of a p-adic formula of Bertolini-Darmon-Prasanna obtained in our earlier work, and an exceptional zero formula for Heegner points. By independent approaches different from ours, a similar p-converse theorem was obtained by Skinner--Zhang under additional ramification hypotheses on Ep, and by Venerucci assuming finiteness of the p-primary part of the Tate-Shafarevich group.