Let $A$ be a supersingular elliptic curve over $\mathbb{Z}/p\mathbb{Z}$ and $\mathcal{O}$ an order in an imaginary quadratic field contained in the quaternion algebra $\operatorname{End}(A)$, then by Deuring's lifting lemma, $A$ lifts to an elliptic curve $E$ over $\bar{\mathbb{Q}}$ with complex multiplication by an order $\mathcal{O}'$ containing $\mathcal{O}$. For a reference, see "supersingular j-invariants as singular moduli mod-p" by M. Kaneko. My question is about whether this sort of assertion about lifting can be made quantitative in the following sense.

Let $\mathcal{S}$ be the set of all imaginary quadratic fields $K=\mathbb{Q}(\sqrt{-D})$, and $\mathcal{S}_A$ the subset of $\mathcal{S}$ consisting of all imaginary quadratic fields $K$ such that there exists an elliptic curve $E$ over $\bar{\mathbb{Q}}$ with complex multiplication by $\mathcal{O}_K$ such that $E$ is a lift of $A$. Given a real number $x>0$, set $\mathcal{S}(x)$ to consist of $K=\mathbb{Q}(\sqrt{-D})$ such that $D$ is positive, squarefree and $D\leq x$; set $\mathcal{S}_A(x):=\mathcal{S}_A\cap \mathcal{S}(x)$. The sets $\mathcal{S}(x)$ and $\mathcal{S}_A(x)$ are finite, set $\#\mathcal{S}(x)$ (resp. $\#\mathcal{S}_A(x)$) to denote the cardinality of $\mathcal{S}(x)$ (resp. $\mathcal{S}_A(x)$). Then, what can be said about the relative upper density $\limsup_{x\rightarrow \infty} \frac{\#\mathcal{S}_A(x)}{\#\mathcal{S}(x)}$? Is it positive?

There are only finitely many elliptic curves $A$ over $\mathbb{Z}/p\mathbb{Z}$, perhaps these densities are the same for all such $A$. This would be some sort of statement about the equi-distibution of CM j invariants with respect to mod-$p$ reduction. Working explicitly with quarterion algebras (as is done in loc. cit.), it seems to be difficult to derive such a statement. However, perhaps there are tools from arithmetic geometry to prove such a statement? In fact, I'm looking for asymptotic lower bounds for $\#\mathcal{S}_A(x)$ as $x\rightarrow \infty$. I'm hoping to do better than simply show that the set $\mathcal{S}_A$ is infinite.

The reason why I'm interested in this sort of statement has to do with the average behavior of the classical Iwasawa $\lambda$-invariant of an imaginary quadratic field. A recent preprint by M. Stokes https://arxiv.org/abs/2302.09594 establishes the connection between the $\lambda$-invariant of the cyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field $K$ in which $p$ splits, and the number of points on the mod-$p$ reduction of any CM elliptic curve $E$ with complex multiplication by $\mathcal{O}_K$.

EDIT: Let $\mathcal{S}$ be the set of all imaginary quadratic fields $K=\mathbb{Q}(\sqrt{-D})$ such that $p$ is inert in $K$. The set $\mathcal{S}(x)$ consists of $K=\mathbb{Q}(\sqrt{-D})$ in $\mathcal{S}$ such that $D$ is positive squarefree and $D\leq x$. This seems to be setting in which this question is nontrivial, although it doesn't quite square with the context in which Stokes' result is proven.

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