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.

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Abstract

Let $C$ be an irreducible projective curve of degree $d$ in $\mathbb{P}^3$ , defined over $\overline{\mathbb{Q}}$ . It is shown that $C$ has $O_{\varepsilon,d}(B^{2/d+\varepsilon})$ rational points of height at most $B$ , for any $\varepsilon>0$ , uniformly for all curves $C$ . This result extends an estimate of Bombieri and Pila [Duke Math. J., 59 (1989), 337-357] to projective curves.

For a projective surface $S$ in $\mathbb{P}^3$ of degree $d\ge 3$ it is shown that there are $O_{\varepsilon,d}(B^{2+\varepsilon})$ rational points of height at most $B$ , of which at most $O_{\varepsilon,d}(B^{52/27+\varepsilon})$ do not lie on a rational line in $S$ . For non-singular surfaces one may reduce the exponent to $4/3+16/9d$ (for $d=4$ or 5) or $\max\{1,3/\sqrt{d}+2/(d-1)\}$ (for $d\ge 6$ ). Even for the surface $x_1^d+x_2^d=x_3^d+x_4^d$ 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 $F(x,y)\in\mathbb{Z}[x,y]$ 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

$\#\{(x_1,x_2,x_3)\in\mathbb{N}^3:x_1^d+x_2^d+x_3^d=N\} \ll_{\varepsilon,d} N^{\theta/d+\varepsilon}$

with

$\theta=\frac{2}{\sqrt{d}}+\frac{2}{d-1}.$

When $d\ge 8$ this provides the first non-trivial bound for the number
of representations as a sum of three $d$ -th powers.

Item Type:	Article
Subjects:	H - N > Number theory
Research Groups:	Number Theory Group
ID Code:	138
Deposited By:	Roger Heath-Brown
Deposited On:	16 Dec 2004
Last Modified:	20 Jul 2009 14:18

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