Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves
Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves
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DOI：
10.4007/annals.2015.181.1.3
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发表时间：
2015-01-01
影响因子：
4.9
通讯作者：
Shankar, Arul
中科院分区：
文献类型：
--
作者：
Bhargava, Manjul;Shankar, Arul
证明了具有有界不变量的二元四次型的渐近数定理；这将高斯和达文波特分别在二次和三次情况下的经典结果推广到四次情况。我们的技术是相当普遍的，可以应用于计算代数群的其他表示中的积分轨道。我们用这些计数结果证明了Q上的椭圆曲线的平均秩，当按其高度排序时，是有界的。特别地，我们证明了当椭圆曲线按高度排序时，2-Selmer群的平均大小为3。这意味着椭圆曲线的平均秩的极限不超过1.5。
We prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants; this extends, to the quartic case, the classical results of Gauss and Davenport in the quadratic and cubic cases, respectively. Our techniques are quite general and may be applied to counting integral orbits in other representations of algebraic groups.We use these counting results to prove that the average rank of elliptic curves over Q, when ordered by their heights, is bounded. In particular, we show that when elliptic curves are ordered by height, the mean size of the 2-Selmer group is 3. This implies that the limsup of the average rank of elliptic curves is at most 1.5.