论文标题
整数可作为两个理性立方的总和
Integers expressible as the sum of two rational cubes
论文作者
论文摘要
我们证明，正整数的积极比例是两个理性立方体的总和，而积极比例的表达不那么表达，从而证明了Davenport的猜想。更普遍地，我们证明，在任何立方扭曲家族中，椭圆曲线的正比例（实际上至少是六个）的等级为0，并且在任何立方扭曲家族中的2个椭圆形曲线的正比例（实际上，至少是六分之一）的椭圆形曲线，其中2个位置为1。 我们的方法涉及证明，在任何具有任何给定的根号的任何立方扭曲家族中，2-纤维的椭圆曲线的平均大小为3。我们通过概括参数化来实现这一目标，这是由于第二个作者和HO，椭圆形曲线具有额外结构的额外结构。然后，我们使用数量的方法和圆形方法的新颖组合，该方法基于露丝（Ruth）和第一作者的早期作品。特别是，我们利用了对相关组的真实和$ p $ -Adic HAAR措施在圆形方法中产生的奇异积分和系列的新解释。我们证明了相关四边形上不可或缺的一点的均匀性估计，这使我们能够证明，两组扭曲家族中2型组的平均大小的平均大小为3。与给定的根数中的曲线中的曲线子集的子集相位，我们将进一步的筛子表明，该筛子又显示了同一均等的均等数字，而该均要均一数字，这是一个平均值，均为curtvive foreve foreve foreve foreve foreve feffer Is cortive yevive survive survive ye curvive，curveve是curd的3个曲线。 $ p $ - dokchitser-dokchitser和burungale-Skinner的$ p $ converse定理的定理将结束。 我们还证明了平方数序列的上述结果的类似物：即，我们证明正方形整数的正比例为两个有理数的总和，而正比例不是。
We prove that a positive proportion of integers are expressible as the sum of two rational cubes, and a positive proportion are not so expressible, thus proving a conjecture of Davenport. More generally, we prove that a positive proportion (in fact, at least one sixth) of elliptic curves in any cubic twist family have rank 0, and a positive proportion (in fact, at least one sixth) of elliptic curves with good reduction at 2 in any cubic twist family have rank 1. Our method involves proving that the average size of the 2-Selmer group of elliptic curves in any cubic twist family, having any given root number, is 3. We accomplish this by generalizing a parametrization, due to the second author and Ho, of elliptic curves with extra structure by pairs of binary cubic forms. We then use a novel combination of geometry-of-numbers methods and the circle method that builds on earlier work of Ruth and the first author. In particular, we make use of a new interpretation of the singular integral and series arising in the circle method in terms of real and $p$-adic Haar measures on the relevant group. We prove a uniformity estimate for integral points on the relevant quadric, which along with a sieve allows us to prove that the average size of the 2-Selmer group over the cubic twist family is 3. By suitably partitioning the subset of curves in the family with given root number, we effect a further sieve to show that the root number is equidistributed and that the same average, now taken over only those curves of given root number, is again 3. Finally, we apply the $p$-parity theorem of Dokchitser-Dokchitser and a $p$-converse theorem of Burungale-Skinner to conclude. We also prove the analogue of the above results for the sequence of square numbers: namely, we prove that a positive proportion of square integers are expressible as the sum of two rational cubes, and a positive proportion are not.
