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

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