The two-variable Heegner point Iwasawa main conjecture

Let $D_\infty/K$ be the anticyclotomic $\mathbb Z_p$-extension, let $R_\infty=R[[\operatorname{Gal}(D_\infty/K)]]$, and let $\mathfrak{Z}_\infty\in \tilde{H}^1_{f,\operatorname{Iw}}(D_\infty,\mathbf{T}^\dagger)$ be the norm-compatible big Heegner-point class. Assume $R$ is regular, and for a finitely generated torsion $R_\infty$-module $M$ define

$$\u000\operatorname{char}(M)=\prod_{\mathfrak{P}}\mathfrak{P}^{\u000\operatorname{length}(M_\mathfrak{P})},$$

where the product runs over height-one primes of R_\u000\infty, while $\u000\operatorname{char}(M)=0$ if $M$ is not torsion.

The two-variable Heegner point main conjecture. Assuming that $R$ is regular,

$$\operatorname{char}\big(\tilde{H}^1_{f,\operatorname{Iw}}(D_\infty,\mathbf{T}^\dagger) / R_\infty\mathfrak{Z}_\infty\big)^2=\operatorname{char}\big(\tilde{H}^2_{f,\operatorname{Iw}}(D_\infty,\mathbf{T}^\dagger)_{\operatorname{tors}}\big),$$

where the subscript $\operatorname{tors}$ denotes the $R_\infty$-torsion submodule.

This conjecture extends Perrin-Riou's Heegner point main conjecture from elliptic curves to a Hida family. Partial results toward Perrin-Riou's original conjecture are known, and the surrounding results establish the expected rank-one structure of the relevant Iwasawa cohomology modules under the stated non-CM hypothesis.