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


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 \(\mathbb {Q}\).

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The discussion with Ashay Burungale and Francesc Castella leads us to improve the main result and the exposition in an earlier version significantly. We deeply thank to both. This research was partially supported by a KIAS Individual Grant (SP054102) via the Center for Mathematical Challenges at Korea Institute for Advanced Study and by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2018R1C1B6007009). We would like to thank the referee for carefully reading our manuscript and for giving constructive and valuable comments.

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Correspondence to Chan-Ho Kim.

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Kim, CH. On the soft p-converse to a theorem of Gross–Zagier and Kolyvagin. Math. Ann. (2022).

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Mathematics Subject Classification

  • 11F67
  • 11G40
  • 11R23