Vol. 15, No. 7, 2021

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

Ashay Burungale, Francesc Castella and Chan-Ho Kim

Vol. 15 (2021), No. 7, 1627–1653
DOI: 10.2140/ant.2021.15.1627
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.

Keywords
Iwasawa theory, Heegner points, Euler systems, $p$-adic $L$-functions
Primary: 11R23
Secondary: 11F33
Milestones
Received: 28 August 2019
Revised: 4 September 2020
Accepted: 12 October 2020
Published: 1 November 2021
Authors
 Ashay Burungale Department of Mathematics California Institute of Technology Pasadena, CA United States Francesc Castella Department of Mathematics University of California Santa Barbara, CA United States Chan-Ho Kim Center for Mathematical Challenges Korea Institute for Advanced Study Seoul South Korea