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Published August 2, 2022 | In Press
Journal Article Open

p∞-Selmer groups and rational points on CM elliptic curves


Let E/Q be a CM elliptic curve and p a prime of good ordinary reduction for E. We show that if Sel_(p∞) (E/Q) has Zₚ-corank one, then E(Q) has a point of infinite order. The non-torsion point arises from a Heegner point, and thus ordₛ₌₁ L(E,s) = 1, yielding a p-converse to a theorem of Gross–Zagier, Kolyvagin, and Rubin in the spirit of [49, 54]. For p > 3, this gives a new proof of the main result of [12], which our approach extends to all primes. The approach generalizes to CM elliptic curves over totally real fields [4].

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© The Author(s) 2022. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Received: 21 July 2021. Published 08 July 2022. To Bernadette Perrin-Riou, with admiration. We thank Matthias Flach, Jacob Ressler and Qiyao Yu for helpful discussions. We also thank Henri Darmon and Antonio Lei for giving us the opportunity to contribute to this special issue, and the anonymous referee for a detailed reading. During the preparation of this paper, A.B. was partially supported by the NSF grant DMS-2001409; F.C. was partially supported by the NSF grants DMS-1946136 and DMS-2101458; C.S. was partially supported by the Simons Investigator Grant #376203 from the Simons Foundation and by the NSF Grant DMS-1901985; Y.T. was partially supported by the NSFC grants #11688101 and #11531008.

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August 22, 2023
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