2024/09/02 by Castella, Francesc
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2409.01360
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 E[p], and by Venerucci assuming finiteness of the p-primary part of the Tate-Shafarevich group.