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

TL;DR

This paper proves Perrin-Riou's Iwasawa main conjecture for Heegner points on elliptic curves over imaginary quadratic fields, utilizing bipartite Euler systems and Kolyvagin's conjecture, with additional results when p splits in K.

Contribution

It provides the first proof of Perrin-Riou's conjecture under mild hypotheses, advancing understanding of the arithmetic of elliptic curves and Heegner points.

Findings

Proof of Perrin-Riou's Heegner point main conjecture.

Verification of the Iwasawa-Greenberg main conjecture when p splits in K.

Application of bipartite Euler systems and Kolyvagin's conjecture techniques.

Abstract

Let $E/\mathbf{Q}$ 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 $\mathbf{Z}_p$-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-Prasanna.