论文标题
计数椭圆形曲线具有规定的水平结构在数字字段上
Counting elliptic curves with prescribed level structures over number fields
论文作者
论文摘要
Harron和Snowden计算了$ \ Mathbb {Q} $上的椭圆曲线的数量，高到高度$ x $，torsion Group $ g $，每个可能的Torsion Group $ g $ cub over $ \ mathbb {q} $。在本文中，我们将它们的结果推广到所有数字字段和所有级别的结构$ g $，以使相应的模块化曲线$ x_g $是加权的投影线$ \ mathbb {p}（w_0，w_1）$和态度$ x_g \ x（1）$满足特定条件。特别是，这包括所有模块化曲线$ x_1（m，n）$，带有粗模量$ 0 $。我们通过在未发表的邓小平上定义$ \ mathbb {p}（w_0，w_1）$上的大小函数来证明我们的结果，并确定如何计算$ \ mathbb {p}（w_0，w_1，w_1）$上的点数$ x $。
Harron and Snowden counted the number of elliptic curves over $\mathbb{Q}$ up to height $X$ with torsion group $G$ for each possible torsion group $G$ over $\mathbb{Q}$. In this paper we generalize their result to all number fields and all level structures $G$ such that the corresponding modular curve $X_G$ is a weighted projective line $\mathbb{P}(w_0,w_1)$ and the morphism $X_G\to X(1)$ satisfies a certain condition. In particular, this includes all modular curves $X_1(m,n)$ with coarse moduli space of genus $0$. We prove our results by defining a size function on $\mathbb{P}(w_0,w_1)$ following unpublished work of Deng, and working out how to count the number of points on $\mathbb{P}(w_0,w_1)$ up to size $X$.
