Vol. 15, No. 7, 2021

Download this article
Download this article For screen
For printing
Recent Issues

Volume 15
Issue 8, 1865–2122
Issue 7, 1593–1864
Issue 6, 1343–1592
Issue 5, 1077–1342
Issue 4, 821–1076
Issue 3, 569–820
Issue 2, 309–567
Issue 1, 1–308

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 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 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 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
Mathematical Subject Classification
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