Michel–Venkatesh adelic mixing conjecture at level $U$

Let $\varphi\in C_c(Y_U\times Y_U)$, let $\{\mathscr D_i\}_i$ be a sequence of homogeneous toral data, and let $s_i\in\mathbf T_i({\mathbb A}_f)$ be shifts. Let $D_i$ be the discriminant of $\mathscr O_{\mathscr D_i}$, and let $q_i$ be the minimal norm of an integral ideal representing the class of $s_i$ in

$$[\mathbf T_i({\mathbb A}_f)]_{K_{\mathscr D_i}}\simeq \operatorname{Pic}(\mathscr O_{\mathscr D_i}).$$

Michel–Venkatesh adelic mixing conjecture. If $|D_i|\to\infty$ and $q_i\to\infty$ as $i\to\infty$, then

$$\mathscr P^\Delta_{\mathscr D_i}(\varphi;s_i)\xrightarrow[i\to\infty]{}\int_{Y_U\times Y_U}\varphi\,\mathrm d(\mu\times\mu).$$

This is the adelic formulation of mixing for joint Heegner periods: as both the discriminants and the complexities of the shifts grow, the shifted joint packets should equidistribute according to the product of the ambient invariant measures. The source presents it as the mixing conjecture of Michel and Venkatesh.