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TITLE A proof of Perrin-Riou's Heegner point main conjecture
KIAS AUTHORS Kim, Chan-Ho,Kim, Chan-Ho
JOURNAL ALGEBRA & NUMBER THEORY, 2021
ARCHIVE https://arxiv.org/abs/1908.09512
ABSTRACT Let E/Q be an elliptic curve of conductor N, let p > 3 be a prime where E has good ordinary reduction, and let K be an imaginary quadratic field satisfying the Heegner hypothesis. In 1987, Perrin-Riou formulated an Iwasawa main conjecture for the Tate-Shafarevich group of E over the anticyclotomic Z(p)-extension of K in terms of Heegner points. In this paper, we give a proof of Perrin-Riou's conjecture under mild hypotheses. Our proof builds on Howard's theory of bipartite Euler systems and Wei Zhang's work on Kolyvagin's conjecture. In the case when p splits in K, we also obtain a proof of the Iwasawa-Greenberg main conjecture for the p-adic L-functions of Bertolini, Darmon and Prasanna.
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