The Density of Rational Points on Curves and Surfaces 论文

摘要

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.\n\nFor 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.\n\nAs 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.\n\nThe results are not restricted to projective surfaces, and as an application of other results in the paper it is shown that\n\n$\\#\\{(x_1,x_2,x_3)\\in\\mathbb{N}^3:x_1^d+x_2^d+x_3^d=N\\}\n\\ll_{\\varepsilon,d} N^{\\theta/d+\\varepsilon}$\n\nwith\n\n$\\theta=\\frac{2}{\\sqrt{d}}+\\frac{2}{d-1}.$\n\nWhen $d\\ge 8$ this provides the first non-trivial bound for the number\nof representations as a sum of three $d$-th powers.