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

Burungale, Ashay; Castella, Francesc; Kim, Chan-Ho

doi:10.2140/ant.2021.15.1627

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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

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

Vol. 15, No. 7, 2021