# The density of rational points on curves and surfaces

Heath-Brown, D. R. (2002) The density of rational points on curves and surfaces. Annals of Mathematics, 155 . pp. 553-559.

 Preview
PDF
306Kb

## Abstract

Let be an irreducible projective curve of degree in , defined over . It is shown that has rational points of height at most , for any , uniformly for all curves . This result extends an estimate of Bombieri and Pila [Duke Math. J., 59 (1989), 337-357] to projective curves.

For a projective surface in of degree it is shown that there are rational points of height at most , of which at most do not lie on a rational line in . For non-singular surfaces one may reduce the exponent to (for or 5) or (for ). Even for the surface this last result improves on the previous best known.

As a further application it is shown that almost all integers represented by an irreducible binary form have essentially only one such representation. This extends a result of Hooley [J. Reine Angew. Math., 226 (1967), 30-87] which concerned cubic forms only.

The results are not restricted to projective surfaces, and as an application of other results in the paper it is shown that

with

When this provides the first non-trivial bound for the number
of representations as a sum of three -th powers.

Item Type: Article H - N > Number theory Number Theory Group 138 Roger Heath-Brown 16 Dec 2004 20 Jul 2009 14:18

Repository Staff Only: item control page