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arXiv · 2109.12344

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

Chan-Ho Kim
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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.

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Chan-Ho Kim. 2021-09-25. On the soft $p$-converse to a theorem of Gross-Zagier and Kolyvagin. https://arxiv.org/abs/2109.12344

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