TY - JOUR

T1 - On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes

AU - Castella, Francesc

AU - Grossi, Giada

AU - Lee, Jaehoon

AU - Skinner, Christopher

N1 - Funding Information:
This paper has its origins in one of the projects proposed by F.C. and C.S. at the 2018 Arizona Winter School on Iwasawa theory, and we would like to thank the organizers for making possible a uniquely stimulating week during which G.G. and J.L. made very substantial progress on this project. We also thank Ashay Burungale and the anonymous referees for many helpful comments and suggestions on an earlier draft of this paper. G.G. is grateful to Princeton University for the hospitality during a visit to F.C. and C.S. in February-March 2019. During the preparation of this paper, F.C. was partially supported by the NSF grant DMS-1946136 and DMS-2101458; G.G. was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London; C.S. was partially supported by the Simons Investigator Grant #376203 from the Simons Foundation and by the NSF grant DMS-1901985.
Funding Information:
This paper has its origins in one of the projects proposed by F.C. and C.S. at the 2018 Arizona Winter School on Iwasawa theory, and we would like to thank the organizers for making possible a uniquely stimulating week during which G.G. and J.L. made very substantial progress on this project. We also thank Ashay Burungale and the anonymous referees for many helpful comments and suggestions on an earlier draft of this paper. G.G. is grateful to Princeton University for the hospitality during a visit to F.C. and C.S. in February-March 2019. During the preparation of this paper, F.C. was partially supported by the NSF grant DMS-1946136 and DMS-2101458; G.G. was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London; C.S. was partially supported by the Simons Investigator Grant #376203 from the Simons Foundation and by the NSF grant DMS-1901985.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2022/2

Y1 - 2022/2

N2 - Let E/ Q be an elliptic curve and p an odd prime where E has good reduction, and assume that E admits a rational p-isogeny. In this paper we study the anticyclotomic Iwasawa theory of E over an imaginary quadratic field in which p splits, which we relate to the anticyclotomic Iwasawa theory of characters by a variation of the method of Greenberg–Vatsal. As a result of our study we obtain proofs (under relatively mild hypotheses) of Perrin-Riou’s Heegner point main conjecture, a p-converse to the theorem of Gross–Zagier and Kolyvagin, and the p-part of the Birch–Swinnerton-Dyer formula in analytic rank 1, for Eisenstein primes p.

AB - Let E/ Q be an elliptic curve and p an odd prime where E has good reduction, and assume that E admits a rational p-isogeny. In this paper we study the anticyclotomic Iwasawa theory of E over an imaginary quadratic field in which p splits, which we relate to the anticyclotomic Iwasawa theory of characters by a variation of the method of Greenberg–Vatsal. As a result of our study we obtain proofs (under relatively mild hypotheses) of Perrin-Riou’s Heegner point main conjecture, a p-converse to the theorem of Gross–Zagier and Kolyvagin, and the p-part of the Birch–Swinnerton-Dyer formula in analytic rank 1, for Eisenstein primes p.

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U2 - 10.1007/s00222-021-01072-y

DO - 10.1007/s00222-021-01072-y

M3 - Article

AN - SCOPUS:85116239550

VL - 227

SP - 517

EP - 580

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 2

ER -