Number Theory Seminar

For an elliptic curve E/Q and a prime p, a celebrated "p-converse" to a theorem of Kolyvagin takes the form of the implication: If the p-Selmer rank of E is one, then a certain Heegner point is non-torsion. (The Gross-Zagier formula then allows one to conclude that E has analytic rank one.) After the pioneering work of Skinner and Wei Zhang, a growing number of results are known in the direction of this p-converse. In this talk, I'll describe the proof of a result in the same spirit in the rank two case conjectured by Darmon and Rotger, in which Heegner points are replaced by certain generalised Kato classes. The talk is partly based on a joint work with M. L. Hsieh.