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The density of rational points in curves and surfaces

D. R. Heath-Brown·J. -L. Colliot-Thélène
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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.

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D. R. Heath-Brown, J. -L. Colliot-Thélène. 2004-05-20. The density of rational points in curves and surfaces. https://arxiv.org/abs/math/0405392

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