论文标题
$ \ mathbb {q} $的等级扭曲图的分类 - 椭圆曲线的同学类别
A classification of isogeny-torsion graphs of $\mathbb{Q}$-isogeny classes of elliptic curves
论文作者
论文摘要
令$ \ mathcal {e} $为$ \ mathbb {q} $ - 在$ \ mathbb {q} $上定义的椭圆曲线的同基因类别。 The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each elliptic curve in the $\mathbb{Q}$-isogeny class $\mathcal{E}$, and an edge for each cyclic $\mathbb{Q}$-isogeny of prime degree between elliptic curves in the isogeny class, with the degree recorded as a label of the 边缘。在本文中，我们将同一基因图定义为一个等级图，此外，我们将每个顶点用扭转亚组的抽象组结构标记在相应的椭圆曲线的$ \ mathbb {q} $上。然后，本文的主要结果是对$ \ mathbb {q} $ - 椭圆曲线定义的所有可能的差异曲线类别的分类。
Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$. The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each elliptic curve in the $\mathbb{Q}$-isogeny class $\mathcal{E}$, and an edge for each cyclic $\mathbb{Q}$-isogeny of prime degree between elliptic curves in the isogeny class, with the degree recorded as a label of the edge. In this paper, we define an isogeny-torsion graph to be an isogeny graph where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over $\mathbb{Q}$ of the corresponding elliptic curve. Then, the main result of the article is a classification of all the possible isogeny-torsion graphs that occur for $\mathbb{Q}$-isogeny classes of elliptic curves defined over the rationals.
