Perrin–Riou–Howard Heegner point main conjecture

Let $E$ be an elliptic curve, let $K_ fty/K$ be the anticyclotomic $\mathbf{Z}_p$-extension, and let $\Lambda=\mathbf{Z}_p\llbracket\operatorname{Gal}(K_\infty/K)\rrbracket$. Let $\mathrm{Sel}_{\mathrm{Gr}}(K,\mathbf{T})$ and $\mathrm{Sel}_{\mathrm{Gr}}(K,\mathbf{A})$ be the corresponding $\Lambda$-adic Greenberg ordinary Selmer groups, and let $\kappa_1^\infty\in\mathrm{Sel}_{\mathrm{Gr}}(K,\mathbf{T})$ be the $\Lambda$-adic Heegner cohomology class. Let $\iota:\Lambda\to\Lambda$ be the involution induced by inversion in $\operatorname{Gal}(K_\infty/K)$, and set

$$X=\operatorname{Hom}_{\mathbf{Z}_p}(\mathrm{Sel}_{\mathrm{Gr}}(K,\mathbf{A}),\mathbf{Q}_p/\mathbf{Z}_p).$$

Perrin–Riou–Howard's Heegner point main conjecture. The $\Lambda$-modules $\mathrm{Sel}_{\mathrm{Gr}}(K,\mathbf{T})$ and $\mathrm{Sel}_{\mathrm{Gr}}(K,\mathbf{A})$ have rank $1$ and corank $1$, respectively. There is a torsion $\Lambda$-module $M_\infty$ such that

$$\operatorname{char}(M_\infty)=\operatorname{char}(M_\infty)^\iota,$$ $$X\sim\Lambda\oplus M_\infty\oplus M_\infty,$$

and

$$\operatorname{char}(M_\infty)=\operatorname{char}\left(\frac{\mathrm{Sel}_{\mathrm{Gr}}(K,\mathbf{T})}{\Lambda\kappa_1^\infty}\right).$$

This is a $\Lambda$-adic refinement of the relation between nontrivial Heegner points, Mordell–Weil rank one, and finiteness of the Tate–Shafarevich group. The conjecture was formulated by Perrin–Riou and Howard; the source provides no resolution status.