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

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.

Iwasawa theory, Heegner points, Euler systems, $p$-adic $L$-functions
Mathematical Subject Classification
Primary: 11R23
Secondary: 11F33
Received: 28 August 2019
Revised: 4 September 2020
Accepted: 12 October 2020
Published: 1 November 2021
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
South Korea