Let $E∕\mathbb{Q}$ 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 ${\mathbb{Z}}_p$-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.

让 $乙∕\mathbb{Q}$ 是导体的椭圆曲线 $N$， 让 $磷>3$ 成为一个主要的地方 $乙$ 有很好的普通还原，让 $钾$是满足 Heegner 假设的虚二次场。1987 年，Perrin-Riou 为 Tate-Shafarevich 群制定了 Iwasawa 主要猜想$乙$ 在反循环学上 ${\mathbb{Z}}_{磷}$-的扩展 $钾$ 就海格纳点而言。

在本文中，我们在温和的假设下给出了 Perrin-Riou 猜想的证明。我们的证明建立在 Howard 的二分欧拉系统理论和 Wei Zhang 对 Kolyvagin 猜想的研究之上。在这种情况下$磷$ 分裂 $钾$，我们还获得了 Iwasawa-Greenberg 主要猜想的证明 $磷$-adic $升$- Bertolini、Darmon 和 Prasanna 的功能。