2-Selmer groups and the Birch–Swinnerton-Dyer Conjecture for the congruent number curves☆
2-Selmer groups and the Birch–Swinnerton-Dyer Conjecture for the congruent number curves☆
复制标题
DOI：
10.1016/j.jnt.2009.01.015
复制
发表时间：
2007-06
影响因子：
0.7
通讯作者：
Robert C. Rhoades
中科院分区：
文献类型：
--
作者：
Robert C. Rhoades
我们采用一种方法来计算整数n的个数，对于该整数n，曲线En：y2=x3− n2 x具有给定大小的2-塞尔默群。罗杰·希斯-布朗的两篇论文也讨论了这个问题。与以前的工作相比，我们的分析集中在限制n的素因子的数量。此外，我们还讨论了计算这些塞尔默群的大小与验证Birch猜想和Swinnerton-Dyer猜想之间的联系。渐近公式的关键成分是勒让德符号的“独立性”评估的一个整数的素因子正好k个素因子。
We take an approach toward counting the number of integers n for which the curve En: y2=x3−n2x has 2-Selmer groups of a given size. This question was also discussed in a pair of papers by Roger Heath-Brown. In contrast to earlier work, our analysis focuses on restricting the number of prime factors of n. Additionally, we discuss the connection between computing the size of these Selmer groups and verifying cases of the Birch and Swinnerton-Dyer Conjecture. The key ingredient for the asymptotic formulae is the “independence” of the Legendre symbol evaluated at the prime divisors of an integer with exactly k prime factors.