Main conjecture for Heegner points

Let Λac\Lambda^{\rm ac} be the anticyclotomic Iwasawa algebra, let T\mathbf{T} be the big Galois representation, and suppose SelGr(T){\rm Sel}_{\rm Gr}(\mathbf{T}) is torsion-free of Λac\Lambda^{\rm ac}-rank one. Via the canonical isomorphism

Q(Λac)ΛacdetΛac1RΓ~f(T)Q(Λac)ΛacSelGr(T)ΛacSelGr(T)ι,Q(\Lambda^{\rm ac})\otimes_{\Lambda^{\rm ac}}{\rm \det}_{\Lambda^{\rm ac}}^{-1}\widetilde{\mathbf{R}\Gamma}_f(\mathbf{T})\cong Q(\Lambda^{\rm ac})\otimes_{\Lambda^{\rm ac}}{\rm Sel}_{\rm Gr}(\mathbf{T})\otimes_{\Lambda^{\rm ac}}{\rm Sel}_{\rm Gr}(\mathbf{T})^\iota,

let z~K\widetilde{\mathfrak{z}}_{K_\infty} be the element mapping to yyy_\infty\otimes y_\infty, where yy_\infty is the Λ\Lambda-adic Heegner class. Main conjecture for Heegner points. The element z~K\widetilde{\mathfrak{z}}_{K_\infty} is a Λac\Lambda^{\rm ac}-basis of detΛac1RΓ~f(T){\rm \det}_{\Lambda^{\rm ac}}^{-1}\widetilde{\mathbf{R}\Gamma}_f(\mathbf{T}). This is the determinant-line formulation of the anticyclotomic main conjecture relating Heegner classes to the Selmer complex; the source gives no resolution.

Sources & referencesView supporting material

Primary source

Francesc Castella and Takamichi Sano, “On refined nonvanishing conjectures by Kurihara and Kolyvagin”, arXiv:2601.14504 (2026).

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