On the soft p-converse to a theorem of Gross–Zagier and Kolyvagin

Abstract

We give a proof of a soft version of the p-converse to a theorem of Gross–Zagier and Kolyvagin for non-CM elliptic curves with good ordinary reduction at \(p >3\) under the irreducibility assumption on the residual representation. In particular, no condition on the conductor is imposed. Combining with the known results, we obtain the equivalence for every elliptic curve E over \(\mathbb {Q}\).