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

Abstract

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 O where the curve itself is defined over Zp. 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 O.