How to calculate the proportion of everywhere locally soluble diagonal hypersurfaces

2020/03/25 by Yoshinori Kanamura, Kanamura, Yoshinori, Yoshinosuke Hirakawa +1
Mathematics · #11D09 #11D25 #11D41 #11D72 #11E76 #14G05 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2003.11426

openalex publication_date 2020/03/25 · openalex created_date 2020/04/03 · openalex updated_date 2026/07/28

Abstract

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 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 ℚ-rational points under a hypothesis on the Brauer-Manin obstruction.

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