We prove an asymptotic formula for the number of ${\rm SL}_3({\mathbbZ})$-equivalence classes of integral ternary cubic forms having boundedinvariants. We use this result to show that the average size of the 3-Selmergroup of all elliptic curves, when ordered by height, is 4. This implies thatthe average rank of all elliptic curves, when ordered by height, is less than1.17. Combining our counting techniques with a recent result of Dokchitser andDokchitser, we prove that a positive proportion of all elliptic curves haverank 0. Assuming the finiteness of the Tate-Shafarevich group, we also showthat a positive proportion of elliptic curves have rank 1. Finally, combiningour counting results with the recent work of Skinner and Urban, we show that apositive proportion of elliptic curves have analytic rank 0; i.e., a positiveproportion of elliptic curves have non-vanishing $L$-function at $s=1$. Itfollows that a positive proportion of all elliptic curves satisfy BSD.
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机译：我们证明了具有有界不变量的整数三元三次形式的$ {\ rm SL} _3（{\ mathbbZ}）$等价类的渐近公式。我们使用此结果显示，按高度排序时，所有椭圆曲线的3-Selmer组的平均大小为4。这意味着，按高度排序时，所有椭圆曲线的平均秩小于1.17。将我们的计数技术与Dokchitser和Dokchitser的最新结果相结合，我们证明所有椭圆曲线的正比例都为0。假设Tate-Shafarevich组的有限性，我们还显示了椭圆曲线的正比例为1。最后，将我们的计数结果与Skinner和Urban的最新工作相结合，我们显示出椭圆曲线的正比例具有解析等级0；即，正比例的椭圆曲线在$ s = 1 $时具有不变的$ L $函数。因此，所有椭圆曲线的正比例都满足BSD。
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