#### Vol. 15, No. 7, 2021

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
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.

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/ant

We have not been able to recognize your IP address 54.174.225.82 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

##### 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