The Density of Rational Points on Curves and Surfaces

Let C be an irreducible projective curve of degree d in ³, defined over \\Q. It is shown that C has O\, ₃ (B^2/d+\) rational points of height at most B, for any \>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. a projective surface S in ³ of degree d\ 3 it is shown that there are O\, ₃ (B^2+\) rational points of height at most B, of which at most O\, ₃ (B^52/27+\) 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 \\\1, 3/+2/ (d-1) \\ (for d\ 6). Even for the surface x₁ᵈ+x₂ᵈ=x₃ᵈ+x₄ᵈ this last result improves on the previous best known. a further application it is shown that almost all integers represented by an irreducible binary form F (x, y) \, 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. results are not restricted to projective surfaces, and as an application of other results in the paper it is shown that\\#\\ (x₁, x₂, x₃) \³: x₁ᵈ+x₂ᵈ+x₃ᵈ=N\\\\, ₃ N^\/d+\\=\2+\2d-1. d\ 8 this provides the first non-trivial bound for the number representations as a sum of three d-th powers.