Prasanna--Venkatesh rationality conjecture for automorphic cohomology periods

Let $M$ be the motive associated with the elliptic curve $A/F$, let $r$ be the Beilinson regulator, and let $\mathfrak{a}$ be the complex vector space identified with the relevant $W_{\mathbb{R}}$-invariant Betti realization. Let $\kappa_\delta$ transfer the invariant-space structure to $\mathfrak{a}$, and let $r_2$ be the number of complex places of $F$. For a Jacquet--Langlands lift $\pi$ of $\Pi$ to $G$, write $\Omega^\pi_{(\varepsilon_{\mathbb R},\underline 1)}$ and $\Omega^\pi_{(\varepsilon_{\mathbb R},-\underline 1)}$ for the lowest- and highest-degree periods, and let $\hat H_{-\underline 1}^\ast$ be the corresponding top exterior form.

Prasanna--Venkatesh conjecture. One has

$$r\left(H^1_{\mathcal M}(M,\mathbb{Q}(1))\right)\otimes_{\mathbb{Q}}\mathbb{C}=\mathfrak{a},$$

and, for every $\varepsilon_{\mathbb R}\in\{\pm1\}^{\Sigma_B^{\mathbb R}}$,

$$\frac{\Omega^\pi_{(\varepsilon_{\mathbb R},\underline 1)}}{\Omega^\pi_{(\varepsilon_{\mathbb R},-\underline 1)}}i^{r_2}\kappa_\delta^{-1}\left(\hat H_{-\underline 1}^\ast\right)\in\bigwedge^{r_2}r\left(H^1_{\mathcal M}(M,\mathbb{Q}(1))\right)^\vee.$$

This is a rationality prediction for the periods arising from the Prasanna--Venkatesh action on automorphic cohomology. The paper presents it as a consequence of their conjectures together with the relation between lowest- and highest-degree Eichler--Shimura morphisms, and it remains conjectural.