The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1
The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1
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椭圆曲线的5-Selmer群的平均大小为6，平均秩小于1
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
A. Shankar
中科院分区：
文献类型：
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作者：
M. Bhargava;A. Shankar
在本文中，我们证明了$\mathbb{Q}$上的椭圆曲线的平均秩，当按高度排序时，小于$1$（实际上小于$.885$）。作为我们方法的结果，我们还证明了$\mathbb{Q}$上的所有椭圆曲线中至少有五分之四的曲线的秩为0或1；此外，事实上至少有五分之一的椭圆曲线的秩为0。证明这些定理的主要内容是确定$\mathbb{Q}$上的$5$-Selmer椭圆曲线群的平均大小；我们证明这个平均尺寸是6美元。另一个关键因素是椭圆曲线根数等分布的一个新的下界；我们证明了在$\mathbb{Q}$上存在密度至少为$55\%$的椭圆曲线族，其根数是等分布的。
In this article, we prove that the average rank of elliptic curves over $\mathbb{Q}$, when ordered by height, is less than $1$ (in fact, less than $.885$). As a consequence of our methods, we also prove that at least four fifths of all elliptic curves over $\mathbb{Q}$ have rank either 0 or 1; furthermore, at least one fifth of all elliptic curves in fact have rank 0. The primary ingredient in the proofs of these theorems is a determination of the average size of the $5$-Selmer group of elliptic curves over $\mathbb{Q}$; we prove that this average size is $6$. Another key ingredient is a new lower bound on the equidistribution of root numbers of elliptic curves; we prove that there is a family of elliptic curves over $\mathbb{Q}$ having density at least $55\%$ for which the root number is equidistributed.