THE DENSITY OF RATIONAL POINTS ON NON-SINGULAR HYPERSURFACES, II

2006/08/07 by T. D. Browning, T. D. BROWNING, D. R. Heath‐Brown +3
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Meromorphic and Entire Functions

paper · doi:10.1112/s0024611506015784

Abstract

For any integers d,n ≥ 2, let X ⊂ ℙⁿ be a non-singular hypersurface of degree d that is defined over the rational numbers. The main result in this paper is a proof that the number of rational points on X which have height at most B is O(B^{n - 1 + ε}), for any ε > 0. The implied constant in this estimate depends at most upon d, ε and n.

Citations

• Rational points of bounded height on Fano varieties

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• The density of rational points on non-singular hypersurfaces, I
• The density of rational points on non-singular hypersurfaces, II
• Counting rational points on hypersurfaces
• Counting rational points on smooth hypersurfaces with high degree
• Counting rational points on cubic hypersurfaces