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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?

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    $\begingroup$ CM theory gives an obstruction to density as follows. Any such $j$ generates an abelian extension of an imaginary quadratic field $K\subset \overline{\mathbf{Q}}$, so if $F$ is the compositum of the finitely many quadratic extensions of $\mathbf{Q}_p$ then $J_p \subset F^{\rm{ab}}$. For $\Gamma_F := {\rm{Gal}}(\overline{\mathbf{Q}}_p/F)$, the closed set of fixed points for the (closure of the) commutator of $\Gamma_F$ is $F^{\rm{ab}}$, so anything outside $F^{\rm{ab}}$ cannot be arbitrarily approximated by such $\sigma(j)$'s. $\endgroup$
    – nfdc23
    Commented Mar 24, 2016 at 5:04

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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.

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    $\begingroup$ This seems not quite right, since the canonical lift admits many p-isogenies to (mostly) distinct elliptic curves, whereas there are only two p-isogenies on the special fiber. $\endgroup$
    – Ari Shnidman
    Commented Mar 24, 2016 at 0:28
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    $\begingroup$ In fact, these ordinary disks all have infinitely many CM j-invariants, if you consider curves with CM by orders of conductor divisible by larger and larger powers of p. $\endgroup$
    – Ari Shnidman
    Commented Mar 24, 2016 at 0:32
  • $\begingroup$ Two remarks: 1) It might make sense to consider the closure within the Berkovich space. 2) I remember of old discussions with Rodolphe Richard and Laurent Fargues that concur to believe that, measure theoretically, the supersingular/ordinary CM-points concentrate at the Berkovich generic points of the supersingular/ordinary disks. I do not remember wether we had a proof or not. $\endgroup$
    – ACL
    Commented Mar 24, 2016 at 12:35
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    $\begingroup$ @ACL: I believe the Galois orbit of an ordinary CM point will converge to the Gauss point of the Berkovich affine line, just like in the case of $p$-adic roots of unity (or of small points). The reason for this is very simple: by Deuring, the discriminant of the minimal equation of an ordinary singular modulus is a $p$-adic unit; we should be able to read the conclusion from this. The supersingular case is the rather more interesting one; there, at least the distribution of the $\mod{p}$ residues is known (Michel, The subconvexity problem for Rankin-Selberg $L$-functions...). $\endgroup$
    – Vesselin Dimitrov
    Commented Mar 25, 2016 at 21:22
  • $\begingroup$ (@ACL: Correction: I was thinking just of the fundamental discriminants case. Anyway I think the ordinary case is a lot easier in this problem, like in Andre-Oort. What I had in mind was something like the argument on page 122 in Bombieri-Gubler, but Serre-Tate theory might take care of the ordinary case. I'd be very much interested to know of a solution in the supersingular case!) $\endgroup$
    – Vesselin Dimitrov
    Commented Mar 25, 2016 at 21:47
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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.

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    $\begingroup$ Very interesting. $\endgroup$
    – Pete L. Clark
    Commented Mar 26, 2016 at 19:57
  • $\begingroup$ Thank you. This is very interesting. The supersingular case is what most interests me anyway. Am I to take away from this that it is an open question which points with supersingular residues are accumulation points? $\endgroup$
    – Michael Griffin
    Commented Apr 5, 2016 at 16:38
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    $\begingroup$ You are welcome. I think it's an open question, yes. $\endgroup$
    – Vesselin Dimitrov
    Commented Apr 5, 2016 at 17:02

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