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arXiv · 2003.11426

How to calculate the proportion of everywhere locally soluble diagonal hypersurfaces

Yoshinori Kanamura·Yoshinosuke Hirakawa
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Abstract

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.

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Yoshinori Kanamura, Yoshinosuke Hirakawa. 2020-03-25. How to calculate the proportion of everywhere locally soluble diagonal hypersurfaces. https://arxiv.org/abs/2003.11426

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