论文标题
CM椭圆曲线：火山，现实和应用，第二部分
CM Elliptic Curves: Volcanoes, Reality and Applications, Part II
论文作者
论文摘要
令$ m \ mid n $为正整数，让$δ$为虚构二次field $ k $中订单的判别。当$δ_k<-4 $时，第一作者确定了近x_0（m，n）\ rightArrow x（1）$的纤维上的封闭点$j_δ$对应于$δ$，并显示地图的所有光纤$ x_1（m，n）\ rightArrow x_0（m，n）$ co x_0（m，n）$ cum y ye $j_δ均为$j_δ。在这里，我们通过解决最困难的情况$Δ_K\ in \ { - 3，-4 \} $来补充这项先前的工作。这些作品提供了计算所需的所有信息，对于每个正整数$ d $，$ e（f）的所有子组[\ propatatorName {tors}] $，其中$ f $是$ d $ $ d $和$ e_ {/f} $的数字字段，都是具有复杂乘法的椭圆形曲线。
Let $M \mid N$ be positive integers, and let $Δ$ be the discriminant of an order in an imaginary quadratic field $K$. When $Δ_K < -4$, the first author determined the fiber of the morphism $X_0(M,N) \rightarrow X(1)$ over the closed point $J_Δ$ corresponding to $Δ$ and showed that all fibers of the map $X_1(M,N) \rightarrow X_0(M,N)$ over $J_Δ$ were connected. Here we complement this prior work by addressing the most difficult cases $Δ_K \in \{-3,-4\}$. 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.
