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

TL;DR

This paper proves a soft $p$-converse to a theorem of Gross-Zagier and Kolyvagin for certain elliptic curves, establishing a criterion linking Mordell-Weil rank, Tate-Shafarevich group finiteness, and analytic rank.

Contribution

It provides a proof of a soft $p$-converse for non-CM elliptic curves with good ordinary reduction at $p > 3$, without restrictions on the conductor.

Findings

Mordell-Weil rank is one if and only if the analytic rank is one.

Tate-Shafarevich group is finite under the same condition.

The result holds for all elliptic curves over the rationals with the specified properties.

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.