CM Elliptic Curves: Volcanoes, Reality and Applications

TL;DR

This paper analyzes the fibers of modular curves over CM points, providing explicit counts and structures crucial for classifying torsion subgroups of CM elliptic curves over number fields.

Contribution

It determines the fibers of certain modular curves over CM points and their connectedness, advancing the understanding of CM elliptic curves' torsion structures.

Findings

Fiber of $X_0(M,N) ightarrow X(1)$ over CM points is explicitly determined.

The fiber of $X_1(M,N) ightarrow X_0(M,N)$ over CM points is shown to be connected.

Results enable computation of torsion subgroups of CM elliptic curves over number fields.

Abstract

For positive integers $M \mid N$ and an order of discriminant $\Delta$ in an imaginary quadratic field $K$ with discriminant $\Delta_K < -4$, we determine the fiber of the morphism $X_0(M,N) \rightarrow X(1)$ over the closed point $J_{\Delta}$ corresponding to $\Delta$. We also show that the fiber of the natural map $X_1(M,N) \rightarrow X_0(M,N)$ over $J_{\Delta}$ is connected. Putting this together we deduce the number of points in the fiber of $X_1(M,N) \rightarrow X(1)$ over $J_{\Delta}$ and their residual degrees. In the continuation of this work with F. Saia, these results will be extended to $\Delta_K \in \{-4,3\}$. 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 (CM).