Global rationality and Shimura reciprocity conjecture for Stark–Heegner cycles

Let $V_p(f)$ be the two-dimensional representation attached to a Bianchi modular form, assume that $V_p(f)|_{G_{F_{\mathfrak p}}}$ is semistable, and let $\mathfrak p$, $K$, $H_{\mathcal C}$, $H_\chi$, $L$, $\mathcal O$, $\mathcal R$, $\mathcal C$, $\Psi$, $\mathcal S_\Psi$, $\mathfrak s_\Psi$, and $\mathfrak s_\chi$ be as in the source. Write $\operatorname{res}_{\mathfrak p}$ for the restriction map from the global semistable Bloch–Kato Selmer group to the local semistable cohomology group. Global rationality conjecture. (i) For every optimal embedding $\Psi\in\operatorname{Emb}(\mathcal O,\mathcal R)$ of conductor $\mathcal C$ relatively prime to $\mathcal N\mathcal D_K$, there exists

$$\mathcal S_\Psi\in\operatorname{Sel}_{\mathrm{st}}\left(H_{\mathcal C},V_p(f)\left(\frac{k+2}{2}\right)\right)$$

with

$$\mathfrak s_\Psi=\operatorname{res}_{\mathfrak p}(\mathcal S_\Psi).$$

(ii) For every $\Psi\in\operatorname{Emb}(\mathcal O,\mathcal R)$ and $\mathfrak a\in\operatorname{Pic}(\mathcal O)$, if $\tau=\operatorname{rec}(\mathfrak a)$, then

$$\operatorname{res}_{\mathfrak p}(\mathcal S_\Psi^\tau)=\mathfrak s_{\mathfrak a\cdot\Psi}.$$

(iii) For every character $\chi:\operatorname{Gal}(H_{\mathcal C}/K)\to\mathbb C^\times$, there exists

$$\mathcal S_\chi\in\operatorname{Sel}_{\mathrm{st}}\left(H_\chi,V_p(f)\left(\frac{k+2}{2}\right)\right)^\chi$$

with $\operatorname{res}_{\mathfrak p}(\mathcal S_\chi)=\mathfrak s_\chi$, where $H_\chi$ is the abelian subextension cut out by $\chi$ and the superscript $\chi$ denotes the $\chi$-isotypical subspace. The conjecture asserts that the constructed local Stark–Heegner classes are globally rational and obey Shimura reciprocity; the source presents it as the main conjecture and gives no resolution.