Point counting on reductions of CM elliptic curves

TL;DR

This paper provides explicit formulas for counting points on reductions of CM elliptic curves and constructs models for certain CM Q-curves, extending previous theoretical results in the field.

Contribution

It introduces new explicit formulas for point counts on CM elliptic curve reductions and models for CM Q-curves, broadening the scope of earlier work.

Findings

Explicit formulas for point counts on CM elliptic curves

Models for CM Q-curves in specific cases

Generalization of prior results by Gross and Stark

Abstract

We give explicit formulas for the number of points on reductions of elliptic curves with complex multiplication by any imaginary quadratic field. We also find models for CM $\mathbf{Q}$-curves in certain cases. This generalizes earlier results of Gross, Stark, and others.