论文标题
CM椭圆曲线：火山，现实和应用
CM Elliptic Curves: Volcanoes, Reality and Applications
论文作者
论文摘要
对于积极的整数，$ m \ mid n $和一个判别$δ$的顺序在虚构的二次字段$ k $中，判别$Δ_k<-4 $，我们确定封闭点$j_δ$ to to $δ$ fo to $Δ$的形态$ x_0 $ x_0（m，n）\ rightarrow x（1）$的纤维。我们还表明，自然地图的光纤$ x_1（m，n）\ rightArrow x_0（m，n）$ iver $j_δ$连接。将其汇总在一起，我们将$ x_1（m，n）\ rightArrow x（1）$ $j_δ$及其剩余学位的纤维中的点数推断出。在使用F. Saia进行这项工作时，这些结果将扩展到\ { - 4,3 \} $中的$Δ_K\。这些工作提供了计算所需的所有信息，对于每个正整数$ d $，所有子组的$ e（f）[\ propatatorName {tors}] $，其中$ f $是$ d $ $ d $和$ e _ {/f} $的数字字段，都是带有复杂乘法（CM）的椭圆形曲线。
For positive integers $M \mid N$ and an order of discriminant $Δ$ in an imaginary quadratic field $K$ with discriminant $Δ_K < -4$, we determine the fiber of the morphism $X_0(M,N) \rightarrow X(1)$ over the closed point $J_Δ$ corresponding to $Δ$. We also show that the fiber of the natural map $X_1(M,N) \rightarrow X_0(M,N)$ over $J_Δ$ is connected. Putting this together we deduce the number of points in the fiber of $X_1(M,N) \rightarrow X(1)$ over $J_Δ$ and their residual degrees. In the continuation of this work with F. Saia, these results will be extended to $Δ_K \in \{-4,3\}$. 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).
