CM $j$-invariants in $p$-adic fields I'm trying to understand the $p$-adic distribution of $j$-invariants for elliptic curves with complex multiplication.
Specifically, suppose $\sigma$ is some embedding $\sigma:\overline{\mathbb Q}\to \overline{\mathbb Q_p}$ from the algebraic numbers to the algebraic closure of the $p$-adic rationals. Let's consider the set $$J_{p}=\{\sigma(j) \ : \  j \ \text{ is the } j\text{-invariant of some CM elliptic curve over } \overline{\mathbb Q}\}.$$
Is $J_p$ dense in any neighborhood in $\overline{\mathbb Q_p}$? Does it help if we restrict to a finite extension of $\mathbb Q_p$, or even just to $\mathbb Q_p$ itself?
 A: All accumulation points of $J_p$ in $\mathbb{C}_p$ are roots of degree two monic equations over $\mathbb{Z}_p$, and their approximants are necessarily supersingular at $p$. Moreover, there exist accumulation points. (There is then a further restriction: the reduction of the point has to be one of the $\approx p/12$ supersingular residues in $\mathbb{F}_{p^2}$. Is this perhaps the only restriction?)
First, extending Pete Clark's remarks, the ordinary CM points have no accumulation point in $\mathbb{C}_p$. (So no, the CM invariants aren't dense in any of the ordinary disks.) This is similar to the corresponding fact about the $p$-adic roots of unity. The analogy here is substantiated by the Serre-Tate theory; cf. Prop. 3.5 in de Jong and Noot's paper Jacobians with complex multiplication. Building upon this, P. Habegger (The Tate-Voloch conjecture in a power of a modular curve, Int. Math. Res. Notices 2014) established much more generally that no algebraic subvariety $V/\mathbb{C}_p$ in a power of the modular curve is $p$-adically approximated by ordinary CM points not lying in $V$. The prototypical $\mathbb{G}_m^r$ case, where the special points are the torsion ones, had been established by Tate and Voloch in the same journal (Linear forms in $p$-adic roots of unity, 1996).
So the question reduces to approximating with supersingular points. These belong to the valuation ring of a quadratic extension of $\mathbb{Q}_p$, hence the claim in my opening paragraph. Habegger's Proposition 2 proves that $0$ is an accumulation point of supersingular CM points, in order to demonstrate that the restriction to ordinary points in his main result is essential. This should work on other examples, though I am unsure exactly which quadratic integral elements over $\mathbb{Z}_p$ may be approximated with Habegger's method. At least this shows the existence of accumulation points.
A: Good question.  I am leaving some first thoughts now; I hope to have a chance to think more about it later.
Because CM elliptic curves have potentially good reduction, the $j$-invariant is an algebraic integer and thus lies in the valuation ring -- i.e., the closed unit disk -- of $\overline{\mathbb{Q}_p}$.  So they are not dense overall.
Now take an ordinary $j$-invariant $\overline{j}$ on the affine line over $\overline{\mathbb{F}_p}$.  It is a result of Deuring that there is exactly one CM $j$-invariant corresponding to an order of conductor prime to $p$ (thanks to Ari Shnidman for this correction) which reduces to $\overline{j}$.  (This is the "canonical lift.")  The preimage of $\overline{j}$ is a $p$-adic disk which has one "prime-to-$p$" CM $j$-invariant.  However, it is also part of Deuring's reduction theory that the disk contains the $j$-invariants of CM elliptic curves whose CM discriminant is a power of $p$ times that of the canonical lift -- and no other CM $j$-invariants.  So these "ordinary disks" do contain infinitely many CM $j$-invariants, contrary to what I said before.  I wonder whether they are dense in the disk.  Note that these $j$-invariants are generating ring class fields which are, I believe, increasingly ramified over $\mathbb{Q}_p$, so if we restricted to CM $j$-invariants lying only in some fixed $p$-adic field we would indeed have only finitely many in each ordinary disk.
