The density of rational points on non-singular hypersurfaces, I

2005/02/11 by T. D. Browning, D. R. Heath‐Brown, Browning, T. D. +1
Mathematics · #11G35 (11P05 #14G05) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.math/0502243

openalex publication_date 2005/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X ⊂ ℙⁿ be a non-singular hypersurface of degree d>1, and let ε>0. This paper is concerned with the conjecture that there are O(B^n-1+ε) rational points on X that have height at most B, in which the implied constant is allowed to depend only upon d, ε and n. In particular this conjecture is shown to hold as soon as d>4. Furthermore, the main ideas in the proof are used to obtain new paucity estimates for certain diophantine equations.

Citations

• Rational points of bounded height on Fano varieties

Related

• THE DENSITY OF RATIONAL POINTS ON NON-SINGULAR HYPERSURFACES, II
• The density of rational points on non-singular hypersurfaces, II
• Counting rational points on hypersurfaces
• Counting rational points on smooth hypersurfaces with high degree
• Density of rational points on manifolds and Diophantine approximation on hypersurfaces