CM Elliptic Curves: Volcanoes, Reality and Applications, Part II

TL;DR

This paper extends the understanding of CM elliptic curves by analyzing special discriminant cases, providing detailed fiber structures of modular curves, and enabling computation of torsion subgroups over number fields.

Contribution

It completes the classification of fibers over CM points for discriminants -3 and -4, advancing the study of CM elliptic curves and their torsion structures.

Findings

Determined fiber structures over special CM points for discriminants -3 and -4.

Connected all fibers of certain modular curve maps over these CM points.

Provided methods to compute torsion subgroups of CM elliptic curves over number fields.

Abstract

Let $M \mid N$ be positive integers, and let $\Delta$ be the discriminant of an order in an imaginary quadratic field $K$. When $\Delta_K < -4$, the first author determined the fiber of the morphism $X_0(M,N) \rightarrow X(1)$ over the closed point $J_{\Delta}$ corresponding to $\Delta$ and showed that all fibers of the map $X_1(M,N) \rightarrow X_0(M,N)$ over $J_{\Delta}$ were connected. Here we complement this prior work by addressing the most difficult cases $\Delta_K \in \{-3,-4\}$. These works provide all the information needed to compute, for each positive integer $d$, all subgroups of $E(F)[\operatorname{tors}]$, where $F$ is a number field of degree $d$ and $E_{/F}$ is an elliptic curve with complex multiplication.