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

Abstract

Let $E∕\mathbb{Q}$ be an elliptic curve of conductor $N$, let 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 ${\mathbb{Z}}_p$-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.