Heegner Point Kolyvagin System and Iwasawa Main Conjecture

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 anticyclotomi...

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Bibliographic Details
Published inActa mathematica Sinica. English series Vol. 37; no. 1; pp. 104 - 120
Main Author Wan, Xin
Format Journal Article
LanguageEnglish
Published Beijing Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 01.01.2021
Springer Nature B.V
Subjects
Curves
Mathematical analysis
Mathematics
Mathematics and Statistics
Theorems
Iwasawa theory
11R23
Heegner
Kolyvagin
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Summary: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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ISSN:1439-8516
1439-7617
DOI:10.1007/s10114-021-8355-7