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

@inproceedings{Kim2021OnT, title={On the \$p\$-converse to a theorem of Gross-Zagier and Kolyvagin}, author={Chan-Ho Kim}, year={2021} }

We give a simple proof of a p-converse theorem to a theorem of Gross–Zagier and Kolyvagin for non-CM elliptic curves with good ordinary reduction at p > 3 under mild hypotheses. In particular, no condition on the conductor is imposed. Combining with the known results, we obtain the equivalence rkZE(Q) = 1,#X(E/Q) < ∞ ⇔ ords=1L(E, s) = 1 for every elliptic curve E over Q. 1. Statement of the main result The aim of this short article is to give a succinct proof of the following p-converse result… Expand

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