Michel–Venkatesh mixing conjecture for joint CM packets

Let $Y$ be a complex modular curve. Let $\mathcal{P}_i\subset Y$ be a sequence of packets of CM points, with each $\mathcal{P}_i$ a principal homogeneous space of $\operatorname{Pic}(\Lambda_i)$, where $\Lambda_i$ is the CM order of its points. For each $i\in\mathbb{N}$, fix $\sigma_i\in\operatorname{Pic}(\Lambda_i)$ and define

$$\mathcal{P}_i^\mathrm{joint}=\left\{\left(z,\sigma_i.z\right)\mid z\in\mathcal{P}_i\right\}\subset Y\times Y.$$

Let $\mu_i^\mathrm{joint}$ be the normalized counting measure supported on $\mathcal{P}_i^\mathrm{joint}$, and set

$$\mathfrak{N}_i=\min_{\substack{\mathfrak{a}\subseteq\Lambda_i\ \operatorname{invertible\ ideal}\\ \mathfrak{a}\in\sigma_i}}\operatorname{Nr}\mathfrak{a}.$$

Mixing conjecture. If $\mathfrak{N}_i\to_{i\to\infty}\infty$, then $\mu_i^\mathrm{joint}$ converges weak-$*$ to $\mathrm{m}_Y\times\mathrm{m}_Y$.

This conjecture asserts asymptotic independence of two CM points related by the class-group element $\sigma_i$. The paper proves it for toral packets under additional hypotheses, including splitting at two fixed primes and the absence of exceptional Landau–Siegel zeros; the conjecture itself is not established in full generality.