The density of rational points in curves and surfaces

2004/05/20 by D. R. Heath-Brown, J. -L. Colliot-Thélène · 4 citations
Mathematics · #math.NT

paper · pdf

published as Ann. of Math. (2), Vol. 155 (2002), no. 2, 553--598 · 46 pages, published version; appendix by J.-L. Colliot-Thélène

arxiv created 2004/05/20 · arxiv updated 2009/12/01

Abstract

Let X be an algebraic variety, defined over the rationals. This paper gives upper bounds for the number of rational points on X, with height at most B, for the case in which X is a curve or a surface. In the latter case one excludes from the counting function those points that lie on lines in the surface. The bounds are uniform for all X of a given degree. They are best possible in the case of curves. As an application it is shown that if F is an irreducible binary form of degree 3 or more then almost all integers represented by F have essentially one such representation.

Cited by

• Serre's question on thin sets in projective space
• A geometric determinant method and geometric dimension growth
• The Hasse principle for random homogeneous polynomials in thin sets
• Algebraic aspects of the polynomial Littlewood-Offord problem

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