On the $j$-invariants of CM-elliptic curves defined over $\mathbb{Z}_p$

Abstract

We characterize the possible reductions modulo $p$ of the $j$-invariants of supersingular elliptic curves which admit complex multiplication by a (potentially non-maximal) order $\mathcal O$ where the curve itself is defined over $\mathbb{Z}_p$. In particular, we show that the collection of possible $j$-invariants as well as some aspects of the distribution depends on which primes divide the discriminant and conductor of the order $\mathcal O$.