Beyond heights: slopes and distribution of rational points

2018/06/29 by Emmanuel Peyre, Peyre, Emmanuel · 4 citations
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1806.11437

openalex publication_date 2018/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The distribution of rational points of bounded height on algebraic varieties is far from uniform. Indeed the points tend to accumulate on thin subsets which are images of non-trivial finite morphisms. The problem is to find a way to characterise the points in these thin subsets. The slopes introduced by Jean-Benoît Bost are a useful tool for this problem. These notes will present several cases in which this approach is fruitful. We shall also describe the notion of locally accumulating subvarieties which arises when one considers rational points of bounded height near a fixed rational point.

Citations

• Rational points on some Fano cubic bundles
• Rational points of bounded height on Fano varieties
• Sur le nombre des points rationnels de hauteur borné des variétés algébriques

Cited by

• Local distribution of rational points in flag varieties
• Integral Diophantine approximation on varieties
• Counting D₄-field extensions by multi-invariants
• The spaces of rational curves on del Pezzo surfaces via conic bundles

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