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 $E/\mathbf{Q}$ 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 $E[p]$, and by Venerucci assuming finiteness of the $p$-primary part of the Tate-Shafarevich group.