Ranks, $2$-Selmer groups, and Tamagawa numbers of elliptic curves with $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$-torsion
Ranks, $2$-Selmer groups, and Tamagawa numbers of elliptic curves with $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$-torsion
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$mathbb{Z}/2mathbb{Z} imes mathbb{Z}/8mathbb{Z}$-torsion 的椭圆曲线的秩、$2$-Selmer 群和玉川数
DOI：
10.2140/obs.2019.2.173
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
Wanlin Li
中科院分区：
文献类型：
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作者：
Stephanie Chan;Jeroen Hanselman;Wanlin Li
在2016年，Balakrishnan-Ho-Kaplan-Spicer-Stein-Weigandt [1]制作了一个按高度排序的Q上椭圆曲线的数据库，他们在其中计算了秩，2-塞尔默群的大小和其他算术不变量。他们观察到，在某一点之后，平均排名似乎随着身高的增加而下降。本文考虑Q上有理挠子群同构于Z/2 Z × Z/8 Z的椭圆曲线族.在GRH和BSD的条件下，我们计算了202，461条参数高度小于103的曲线中92%的排名。我们还计算了2-塞尔默群和玉川积的大小，并证明了它们的平均值趋于无穷大。
In 2016, Balakrishnan–Ho–Kaplan–Spicer–Stein–Weigandt [1] produced a database of elliptic curves over Q ordered by height in which they computed the rank, the size of the 2-Selmer group, and other arithmetic invariants. They observed that after a certain point, the average rank seemed to decrease as the height increased. Here we consider the family of elliptic curves over Q whose rational torsion subgroup is isomorphic to Z/2Z× Z/8Z. Conditional on GRH and BSD, we compute the rank of 92% of the 202,461 curves with parameter height less than 103. We also compute the size of the 2-Selmer group and the Tamagawa product, and prove that their averages tend to infinity for this family.
按 Selmer 组和等级的高度和分布排序的椭圆曲线数据库
影响因子：
--
作者：
Balakrishnan J
通讯作者：
Balakrishnan J