论文标题
$ \ mathbb {q} [i] $带有固定扭转的椭圆曲线的频率
The Frequency of Elliptic Curves Over $\mathbb{Q}[i]$ with Fixed Torsion
论文作者
论文摘要
Mazur的定理指出，在合理数字上定义的椭圆曲线的扭转亚组完全存在15个可能性。以前是哈伦和斯诺登表明的，具有特定扭力子组$ g $的椭圆形曲线的同构曲线数量为$ x^{1/{d（g）}} $，对于某些正$ d（g）$，根据$ g $。我们在$ \ mathbb {q} [i] $上计算这些组的$ d（g）$。此外，在最近的一批论文中，证明了基本场中的扭转子组还有9个可能性，$ \ mathbb {q} [i] $。我们计算这些新组的$ d（g）$的价值。
Mazur's Theorem states that there are precisely 15 possibilities for the torsion subgroup of an elliptic curve defined over the rational numbers. It was previously shown by Harron and Snowden that the number of isomorphism classes of elliptic curves of height up to $X$ that have a specific torsion subgroup $G$ is on the order of $X^{1/{d(G)}}$, for some positive $d(G)$ depending on $G$. We compute $d(G)$ for these groups over $\mathbb{Q}[i]$. Furthermore, in a collection of recent papers it was proven that there are 9 more possibilities for the torsion subgroup in the base field $\mathbb{Q}[i]$. We compute the value of $d(G)$ for these new groups.
