The Density of Rational Points on Curves and Surfaces

Let C be an irreducible projective curve of degree d in ³, defined over \̄. It is shown that C has O\ε,d(B2/d+\ε) rational points of height at most B, for any \ε>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\ε,d(B2+\ε) rational points of height at most B, of which at most O\ε,d(B52/27+\ε) 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/\√d+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)\∈[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. 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\\\ε,d N\θ/d+\ε\θ=\2\√d+\2/d-1. d\≥ 8 this provides the first non-trivial bound for the number representations as a sum of three d-th powers.