On The $j$-Invariants of CM-Elliptic Curves Defined Over $\mathbb{Z}_p$

Abstract

We characterize the possible reductions of $j$-invariants of elliptic curves which admit complex multiplication by an order $\mathcal{O}$ where the curve itself is defined over $\mathbb{Z}_p$. In particular, we show that the distribution of these $j$-invariants depends on which primes divide the discriminant and conductor of the order.