The characteristic-ideal equality for generalized Heegner-cycle Selmer modules

Let $(f,K,\mathfrak p)$ be an admissible triple. Let $K_\infty$ be the anticyclotomic $\mathds Z_p$-extension of $K$, let $\Lambda={\mathcal O}_\mathfrak p[[\Gamma_\infty]]$ be its Iwasawa algebra, and let $T$ be the associated $G_\mathds Q$-representation. Write $\hat H^1_f(K_\infty,T)$ for the pro-$p$ Bloch--Kato Selmer group and let $\mathcal H_\infty$ be the $\Lambda$-submodule generated by generalized Heegner cycles. Let $M$ be the finitely generated torsion $\Lambda$-module from the main theorem, so that $\mathcal X_\infty\sim\Lambda\oplus M\oplus M$ and $\operatorname{char}(M)=\operatorname{char}(M)^\iota$. Characteristic-ideal equality conjecture.

$$\operatorname{char}(M)=\operatorname{char}\Big(\hat H^1_f(K_\infty,T)/\mathcal H_\infty\Big).$$

The theorem preceding this conjecture proves only the divisibility of the left-hand characteristic ideal by the right-hand one. The conjecture predicts that the generalized Heegner-cycle submodule accounts exactly for the remaining characteristic ideal, refining the structural description of the Bloch--Kato Selmer group over the anticyclotomic extension.