CM Elliptic Curves: Volcanoes, Reality and Applications

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).