An elliptic curve over a finite field, $k$, of characteristic p is called supersingular if it has no $p$-torsion over $k^{\mathrm{alg}}$, or equivalently, if $\mathrm{End}(E)$ is an order in a quaternion algebra. There are also other equivalent definitions (see, for example Silverman I). These curves are quite rare, since there are 'roughly' only $\frac{p-1}{12}$ of them (see this question for a nice discussion of the count).

Given an elliptic curve $E/K$, where $K$ is a number field, one can ask about the set of primes, $\mathfrak{p}$ of $K$, for which $E$ has supersingular reduction at $\mathfrak{p}$. If $E$ has CM then half of the primes give supersingular reduction (Deuring 1941), but Serre showed that in the non-CM case the primes of supersingular reduction have density zero.

Nonetheless, Elkies showed that for an elliptic curve $E/\mathbb{Q}$, there are infinitely many primes for which the reduction is supersingular (Inv. Math. 1987, Volume 89, Issue 3, pp 561-567). He later extended the argument to $E/K$ where $K$ is any number field with at least one real place. It seems the Elkies argument cannot be easily adapted to curves over a totally imaginary number field.

My question is: what is the current status of this problem? More specifically:

Let $E/K$ be an elliptic curve over a totally imaginary number field. What is known about the number of primes of $K$ for which E has supersingular reduction? Is is known to be infinite?

I also wonder what heuristic would lead one to try to prove that there are infinitely many primes of supersingular reduction:

Given the seeming paucity of supersingular curves, why would one expect $E/K$ to have infinitely many primes of supersingular reduction?


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