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

TL;DR

This paper characterizes the reductions of $j$-invariants of CM elliptic curves over $Z_p$, revealing how their distribution depends on primes dividing the order's discriminant and conductor.

Contribution

It provides a detailed description of the possible reductions of $j$-invariants for CM elliptic curves over $Z_p$, linking their distribution to arithmetic properties of the order.

Findings

Distribution depends on primes dividing discriminant and conductor

Characterization of possible reductions over $Z_p$

Insights into CM elliptic curves' arithmetic properties

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.