论文标题
如何计算局部可溶性对角线超曲面的比例
How to calculate the proportion of everywhere locally soluble diagonal hypersurfaces
论文作者
论文摘要
在本文中，我们建立了一种计算$ \ mathbb {p}^{n} $ of固定度的$ \ mathbb {p}^{n} $的局部可溶性对角线的比例的策略。我们的策略基于Bright，Browning和Loughran建立的产品公式。它们的公式将问题降低为$ \ mathbb {q} _ {v} $的比例的计算 - 所有位置$ v $的可溶性对角线超曲面。作为工作示例，我们在二次和立方体曲面的情况下执行了策略。结果，我们证明，对角立方的$ 99.99 \％$ 4 $ -folds具有$ \ mathbb {q} $ - 在Brauer-Manin障碍物上的假设下的有理点。
In this paper, we establish a strategy for the calculation of the proportion of everywhere locally soluble diagonal hypersurfaces of $\mathbb{P}^{n}$ of fixed degree. Our strategy is based on the product formula established by Bright, Browning and Loughran. Their formula reduces the problem into the calculation of the proportions of $\mathbb{Q}_{v}$-soluble diagonal hypersurfaces for all places $v$. As worked examples, we carry out our strategy in the cases of quadratic and cubic hypersurfaces. As a consequence, we prove that around $99.99\%$ of diagonal cubic $4$-folds have $\mathbb{Q}$-rational points under a hypothesis on the Brauer-Manin obstruction.
