A proof of Perrin-Riou's Heegner point main conjecture

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 Zp-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.