Algebraic-rank criterion and exterior-power description of plectic Heegner classes

Let $A$ be the abelian variety, $E$ the number field, $L_{\mathfrak c}$ the anticyclotomic extension, and let $\chi$ be a character. Let $r$ be the number of primes in $\mathcal{S}$, let $\kappa^{\chi}_{A,\mathcal{S}}$ be the plectic Heegner class, and write $r_\mathrm{alg}(A/E,\chi)$ for the algebraic rank. Let

$$\operatorname{det}\colon \bigwedge^r A(L_{\mathfrak c})\longrightarrow \mathrm{H}^1_f(E_{\otimes,\mathcal{S}},V_p(A))\otimes \mathrm{H}^1_f(L_{\mathfrak c},V_p(A))$$

be the homomorphism defined using the chosen embeddings and the Kummer map. Algebraic-rank criterion for plectic Heegner classes. If $r_\mathrm{alg}(A/E,\chi)\ge r$, then

$$\kappa^{\chi}_{A,\mathcal{S}}\ne 0\quad\iff\quad r_\mathrm{alg}(A/E,\chi)=r.$$

Moreover, there exists $w_{A,\mathcal{S}}^\chi\in\bigwedge^r A(L_{\mathfrak c})^\chi$ such that

$$\kappa^{\chi}_{A,\mathcal{S}}=\operatorname{det}\bigl(w_{A,\mathcal{S}}^\chi\bigr).$$

This conjecture predicts that, once the algebraic rank is at least the number of auxiliary primes, nonvanishing occurs exactly in the minimal-rank case and that the class is generated by an exterior-power element.