The distribution of the Tamagawa ratio in the family of elliptic curves with a two-torsion point
The distribution of the Tamagawa ratio in the family of elliptic curves with a two-torsion point
复制标题
双扭点椭圆曲线族中玉川比的分布
DOI：
10.1186/s40687-014-0015-4
复制
发表时间：
2014
影响因子：
1.2
通讯作者：
Robert J. Lemke Oliver
中科院分区：
文献类型：
--
作者：
Z. Klagsbrun;Robert J. Lemke Oliver
在最近的工作中，Bhargava 和 Shankar 证明了椭圆曲线族中 2-Selmer 群的平均大小恰好为 3，Bhargava 和 Ho 证明了带有标记点的椭圆曲线族中 2-Selmer 群的平均大小恰好为 6。与这些结果相反，我们表明带有两个扭转点的椭圆曲线族中 2-Selmer 群的平均大小为无界。特别是，二扭点的存在意味着理性同构的存在。附加到一对同源曲线的基本量是玉川比率，它测量与同源及其对偶相关的塞尔默群的相对大小。基于之前我们考虑二次扭曲族中的 Tamakawa 比率的工作，我们表明，在所有具有双扭点的椭圆曲线族中，Tamakawa 比率本质上受均值为零且方差不断增长的正态分布控制。
In recent work, Bhargava and Shankar have shown that the average size of the 2-Selmer group of an elliptic curve over is exactly 3, and Bhargava and Ho have shown that the average size of the 2-Selmer group in the family of elliptic curves with a marked point is exactly 6. In contrast to these results, we show that the average size of the 2-Selmer group in the family of elliptic curves with a two-torsion point is unbounded. In particular, the existence of a two-torsion point implies the existence of rational isogeny. A fundamental quantity attached to a pair of isogenous curves is the Tamagawa ratio, which measures the relative sizes of the Selmer groups associated to the isogeny and its dual. Building on previous work in which we considered the Tamagawa ratio in quadratic twist families, we show that, in the family of all elliptic curves with a two-torsion point, the Tamagawa ratio is essentially governed by a normal distribution with mean zero and growing variance.