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

Abstract

Let $E/\mathbf{Q}$ 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 ${\mathbf{Z}}_p$-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-Prasanna.

arXiv comment22 pages. Revised version, to appear in Algebra & Number Theory

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Cited by in corpus (7)

• On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes
• Statistics for Anticyclotomic Iwasawa Invariants of Elliptic Curves
• $p$-adic L-functions via local-global interpolation: the case of ${\mathrm{GL}}_2\times \mathrm{GU}(1)$
• Indivisibility of Heegner cycles over Shimura curves and Selmer groups
• The Iwasawa Main Conjectures for ${\mathrm{GL}}_2$ and derivatives of $p$-adic $L$-functions
• Kolyvagin's Conjecture and patched Euler systems in anticyclotomic Iwasawa theory
• A proof of Kolyvagin's Conjecture via the BDP main conjecture