AI 中文总结
该研究对特征零域上End_K E≅ℤ的椭圆曲线的同源图进行分类，分解问题为p-主同源图分类，引入p- blooming不变量，还应用于刻画潜在复乘椭圆曲线的同源图等场景。
AI 中文摘要
对于在特征零域K上定义且满足End_K E ≅ ℤ的椭圆曲线E，我们对其可能出现的同源图𝒢(E/K)进行分类。我们首先证明，𝒢(E/K)可分解为其素p的p-主同源图的弱笛卡尔积，从而将问题简化为对p-主同源图的分类。随后我们证明，每个此类图作为带边权图，同构于显式带边权图族𝒢_{p^k}^r和𝒢_{p^{∞,+}}^r中的一个成员；除k≥2时的𝒢_{2^k}^0外，其余所有成员均可作为p-主同源图出现。该证明依赖于对E所附p进伽罗瓦表示的详细研究，通过该研究我们将每个图与GL₂(ℤₚ)的一个子群对应起来。更一般地，我们对每个可能的同源图将其与GL₂(ℤ̂)的子群对应，并描述其对应的模曲线，亏格为0的情形下，通过同源椭圆曲线族的参数化完成了同源图的显式参数化。我们还引入了p- blooming不变量ₚ(E/K)，这是一种确定p-主同源图中r值的同源类不变量，并证明具有实嵌入的域上的椭圆曲线可取得最小可能值。作为应用，我们刻画了具有潜在复乘的椭圆曲线的同源图；给出了从adelic伽罗瓦表示确定同源图的算法；重新得到了有理同源图的分类；并在GRH下，对某些数域上出现的同源图进行了分类。
英文摘要
For an elliptic curve $E$ defined over a field $K$ of characteristic $0$ with $\operatorname{End}_K \! E \cong \mathbb{Z}$, we classify which isogeny graphs $\mathcal{G}(E/K)$ can occur. We first show that $\mathcal{G}(E/K)$ decomposes as a weak Cartesian product of its $p$-primary isogeny graphs, one for each prime $p$, thereby reducing the problem to classifying $p$-primary isogeny graphs. We then show that each such graph is isomorphic, as an edge-weighted graph, to a member of an explicit family of edge-weighted graphs $\mathcal{H}_{p^k}^r$ and $\mathcal{H}_{p^{\infty,+}}^r$, every member of which occurs as a $p$-primary isogeny graph except for $\mathcal{H}_{2^k}^0$ for $k\ge 2$. The proof relies on a detailed study of the $p$-adic Galois representation attached to $E$, through which we identify each graph with a subgroup of $\operatorname*{GL}\nolimits_{2}(\mathbb{Z}_{p})$. More generally, we identify subgroups of $\operatorname*{GL}\nolimits_{2}(\widehat{\mathbb{Z}})$ for each possible isogeny graph and describe their corresponding modular curves, completing, in the genus $0$ case, the explicit parameterization of isogeny graphs via parameterized isogenous families of elliptic curves. We also introduce the $p$-blooming invariant $\mathfrak{I}_p(E/K)$, an isogeny class invariant determining the value of $r$ in the $p$-primary isogeny graph, and show that elliptic curves over fields with a real embedding attain the smallest possible value. As applications, we characterize the isogeny graphs of elliptic curves with potential complex multiplication; give an algorithm for determining the isogeny graph from the adelic Galois representation; recover the classification of rational isogeny graphs; and, under GRH, classify the isogeny graphs occurring over certain number fields.
Comments101 pages
