In this paper, we establish a strategy for the calculation of the proportion of everywhere locally soluble diagonal hypersurfaces of ℙn 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 ℚv-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 fourfolds have ℚ-rational points under a hypothesis on the Brauer–Manin obstruction.

中文翻译：

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如何计算无处不在的局部可溶对角超曲面的比例

在本文中，我们建立了一种计算处处局部可溶对角超曲面比例的策略ℙn固定度数。我们的战略基于 Bright、Browning 和 Loughran 建立的产品配方。他们的公式将问题简化为计算比例ℚv-所有地方的可溶对角超曲面v. 作为工作示例，我们在二次和三次超曲面的情况下执行我们的策略。因此，我们证明了99.99%的对角立方四倍有ℚ-Brauer-Manin 阻塞假设下的合理点。