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

By Ashay A. Burungale, Francesc Castella, Chan-Ho Kim. Algebra & Number Theory, 2019

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Let $E/\mathbf{Q}$ be an elliptic curve of conductor $N$, let $p$ be a prime of a good ordinary reduction for $E$, and let $K$ be an imaginary quadratic field satisfying the Heegner hypothesis relative to $N$. 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 a module of Heegner points over the anticyclotomic tower. In this paper, we upgrade Howard's divisibility towards the Heegner point main conjecture to a proof of the predicted equality. Our method builds on W. Zhang's results on Kolyvagin's conjecture, combined with the $p$-adic Waldspurger formula and Kolyvagin's structure theorem for Tate-Shafarevich groups in the rank one case, and with Howard's theory of bipartite Euler systems in the general rank case.

Subfields: Geometry and Topology, History and Philosophy of Science, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b2e255cc47c06090d96e44d0dac7951e9e5f9088