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 that the Mordell-Weil rank is one and the Tate-Shafarevich group is finite if and only if the analytic rank is one for every elliptic curve over the rationals.

arXiv commentTitle changed. The revision is made following the referee's suggestions