CM Elliptic Curves: Volcanoes, Reality and Applications

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 $X_0(M,N)\to X(1)$ over the closed point $J_{\Delta }$ corresponding to $\Delta$. We also show that the fiber of the natural map $X_1(M,N)\to X_0(M,N)$ over $J_{\Delta }$ is connected. Putting this together we deduce the number of points in the fiber of $X_1(M,N)\to X(1)$ 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 $E(F)[\mathrm{tors}]$, where $F$ is a number field of degree $d$ and $E_{/F}$ is an elliptic curve with complex multiplication (CM).

105 pages