Abstract
We prove an anticyclotomic Iwasawa main conjecture proposed by Perrin-Riou for Heegner points for semi-stable elliptic curves E over a quadratic imaginary field K satisfying a certain generalized Heegner hypothesis, at an ordinary prime p. It states that the square of the index of the anticyclotomic family of Heegner points in E equals the characteristic ideal of the torsion part of its Bloch-Kato Selmer group (see Theorem 1.3 for precise statement). As a byproduct we also prove the equality in the Greenberg-Iwasawa main conjecture for certain Rankin-Selberg product (Theorem 1.7) under some local conditions, and an improvement of Skinner’s result on a converse of Gross-Zagier and Kolyvagin theorem (Corollary 1.11).
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Acknowledgements
We would like to thank A. Burungale, B. Howard, M-L Hsieh, C. Skinner and W. Zhang for useful communications. We also thank anonymous referees for many useful suggestions in improving the paper.
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Supported by the Chinese Academy of Science (Grant No. Y729025EE1), NSFC (Grant Nos. 11688101, 11621061) and an NSFC grant associated to the recruitment Program of Global Experts
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Wan, X. Heegner Point Kolyvagin System and Iwasawa Main Conjecture. Acta. Math. Sin.-English Ser. 37, 104–120 (2021). https://doi.org/10.1007/s10114-021-8355-7
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DOI: https://doi.org/10.1007/s10114-021-8355-7