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 - 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

SN - 0020-9910

VL - 227

SP - 517

EP - 580

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 2

ER -