论文标题
显式分类有理椭圆形曲线
Explicit classification of isogeny graphs of rational elliptic curves
论文作者
论文摘要
令$ n> 1 $为一个整数，以便$ x_ {0} \！\ left（n \右）$具有属$ 0 $，而让$ k $为特征$ 0 $或相对较高至$ 6n $的字段。在本文中，我们明确地对所有有理椭圆曲线的等级图进行了分类，这些曲线接受了$ \ m athbb {q} $的非平凡的同学。我们通过介绍$ 56 $的椭圆曲线的$ 56 $参数化的家族$ \ MATHCAL {C} _ {n，i}（t，d，d）$在$ k（t，d）$上定义的$，在$ k（t，d）$上定义了以下两个属性$ n $的属性：固定的两个属性：椭圆曲线$ \ nathcal \ nathcal {c} is { $ k（t，d）$，并且有整数$ k_ {1} $和$ k_ {2} $，使得$ j $ -invariants的$ \ Mathcal {c} _ {c} _ {n，k_ {1}}}}（t，t，d）$ and $ \ mathcal {c}弗里克参数化。结果，我们表明，如果$ e $是数字字段$ k $的椭圆曲线，而等值级别的班级程度可将$ n \ in \ in \ weft \ weft \ {4,6,9 \ right \} $排除，那么$ e $的$ $ e $的二次转换是可以准确的$ \ nmatfrak $ \ mathfrak $ \ k $ n $。
Let $n>1$ be an integer such that $X_{0}\!\left( n\right) $ has genus $0$, and let $K$ be a field of characteristic $0$ or relatively prime to $6n$. In this article, we explicitly classify the isogeny graphs of all rational elliptic curves that admit a non-trivial isogeny over $\mathbb{Q}$. We achieve this by introducing $56$ parameterized families of elliptic curves $\mathcal{C}_{n,i}(t,d)$ defined over $K(t,d)$, which have the following two properties for a fixed $n$: the elliptic curves $\mathcal{C}_{n,i}(t,d)$ are isogenous over $K(t,d)$, and there are integers $k_{1}$ and $k_{2}$ such that the $j$-invariants of $\mathcal{C}_{n,k_{1}}(t,d)$ and $\mathcal{C}_{n,k_{2}}(t,d)$ are given by the Fricke parameterizations. As a consequence, we show that if $E$ is an elliptic curve over a number field $K$ with isogeny class degree divisible by $n\in\left\{4,6,9\right\} $, then there is a quadratic twist of $E$ that is semistable at all primes $\mathfrak{p}$ of $K$ such that $\mathfrak{p}\nmid n$.
