Bertolini–Darmon anticyclotomic formula for the theta element and Heegner class

Consider the setup of the anticyclotomic family in the paper and let $\widetilde r=\dim A^{\dagger}(K)_{\mathbf Q}$. If $\widetilde\varepsilon=+1$, then $\widetilde r$ is even and the stated leading-term identity for $\Theta$ holds; if $\widetilde\varepsilon=-1$, then $\widetilde r$ is odd and the stated leading-term identity for $\mathscr P$ holds, with the common square-index factor $\widetilde i(A,f)$. Bertolini–Darmon conjecture. In the $+1$ case, $\Theta$ vanishes at the trivial character to order at least $\widetilde r/2$, $\widetilde I(A,f)=\widetilde i(A,f)^2$, and, up to sign,

$$(\mathrm d^-)^{\widetilde r/2}\Theta(\mathbf 1)=\widetilde i(A,f)\,\operatorname{pf}^-(A).$$

In the $-1$ case, $\mathscr P$ vanishes to order at least $(\widetilde r-1)/2$, $\widetilde I(A,f)=\widetilde i(A,f)^2$, and, up to sign,

$$(\mathrm d^-)^{(\widetilde r-1)/2}\mathscr P(\mathbf 1)=\widetilde i(A,f)\,\operatorname{Pf}^-(A).$$

The identities lie in the modules specified in the source. This is the anticyclotomic refinement related to Bertolini–Darmon; the supplied passage gives no general resolution status.