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The Heegner Point Main Conjecture

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Let E/\bQE/\bQ be an elliptic curve, let p>2p>2 be a prime of good ordinary reduction, and let KK be an imaginary quadratic field satisfying the Heegner hypothesis. Let Λ\Lambda be the anticyclotomic Iwasawa algebra, let SS be the inverse limit of the relevant pp-adic Selmer groups over the anticyclotomic tower, let XX be the Pontryagin dual of the direct-limit p∞p^\infty-Selmer group, and let κ\Heeg∈S\kappa^\Heeg\in S be the Heegner-point Kolyvagin class. Heegner Point Main Conjecture. Both SS and XX have Λ\Lambda-rank one, and

char⁡Λ(X\tors)=char⁡Λ(S/Λ⋅κ\Heeg)2,\operatorname{char}_\Lambda(X_{\tors})=\operatorname{char}_\Lambda(S/\Lambda\cdot\kappa^\Heeg)^2,

where X\torsX_{\tors} is the Λ\Lambda-torsion submodule of XX. This conjecture is a central Iwasawa-theoretic input for pp-converse theorems. The source states that, under its relevant assumption, it is now a theorem in the Eisenstein case, while the formulation itself is presented as the Heegner Point Main Conjecture.

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Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Heegner point main conjecture

    Let E/QE/\mathbb{Q} have good ordinary reduction at pp, let K∞/KK_\infty/K be the anticyclotomic Zp\mathbb{Z}_p-extension, let Λ\Lambda be its Iwasawa algebra, and let H~f1(K,T)\widetilde H_f^1(K,\mathbb{T}) be the cohomology of the Greenberg Selmer complex. Let z∞z_\infty be the normalized system of regularized Heegner points, viewed in H~f1(K,T)\widetilde H_f^1(K,\mathbb{T}). Heegner point main conjecture. The Λ\Lambda-rank of H~f1(K,T)\widetilde H_f^1(K,\mathbb{T}) is one, and there is a Λ\Lambda-basis

    z~∞∈det⁡Λ−1(RΓf(K,T))\widetilde{\mathfrak{z}}_\infty\in\det_\Lambda^{-1}(\mathbf{R}\Gamma_f(K,\mathbb{T}))

    such that its image under the canonical determinant isomorphism is z∞⊗z∞z_\infty\otimes z_\infty. This predicts that the determinant class of the Selmer complex is generated by the square of the universal Heegner point system; the source gives no resolution.

    source: Takamichi Sano, “On the Tamagawa number conjecture for modular forms twisted by anticyclotomic Hecke characters”, arXiv:2510.01601 (2025).

References

Primary source

Timo Keller and Mulun Yin, “p-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes”, arXiv:2410.23241 (2024).

elliptic-curvesheegner-pointsiwasawa-theory
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