Algebraicity conjecture for plectic Stark–Heegner points

Let $F$ be a totally real number field, let $E/F$ be a quadratic extension, let $A/E$ be the abelian variety in the paper, and let $S=\{\mathfrak p_1,\ldots,\mathfrak p_r\}$ be the relevant set of primes. Write $A(E)_{\mathbb Q}=A(E)\otimes_{\mathbb Z}\mathbb Q$, let $\bigwedge^r(A(E)_{\mathbb Q})$ denote its $r$-th exterior power, and let $\operatorname{det}$ be the determinant map from this exterior power to $\widehat{A}(E_S)_{\mathbb Q}$. The plectic Stark–Heegner point is $\mathrm{P}_{A,S}\in\widehat{A}(E_S)_{\mathbb Q}$, and $r_{\mathrm{alg}}(A/E)$ denotes the Mordell–Weil rank.

Algebraicity conjecture. If $r_{\mathrm{alg}}(A/E)\geq r$, then there exists $w_{A,S}\in\bigwedge^r(A(E)_{\mathbb Q})$ such that

$$\mathrm{P}_{A,S}=\operatorname{det}(w_{A,S}).$$

Moreover, if $\mathrm{P}_{A,S}\neq 0$, then $r_{\mathrm{alg}}(A/E)=r$.

This conjecture predicts that plectic Stark–Heegner points arise algebraically from rational points on $A$. The case $r=1$ follows from the Gross–Zagier–Kolyvagin theorem for totally real fields, while the general case is not established in the supplied text.