The density of rational points on curves and surfaces

By D. R. Heath-Brown, J.-L. Colliot-Th'elene. 2002

262 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Information Systems

View Paper: https://www.semanticscholar.org/paper/4f3ba676bb886e2677d4aabeb67b9a9e7519c92b

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Citations in the corpus

The rational points of a definable set (2006, methodology): “A similar result is achieved by p-adic means in the algebraic setting in [13].”
The Density of Rational Points on Non‐Singular Hypersurfaces, I (2005, background): “With this in mind, we define the counting function M(f ;B) = #{t ∈ Zν : f(t) = 0, |t| B}, (1.2) for any B 1.”
Integral points on elliptic curves and 3-torsion in class groups (2004, methodology): “if we proceeded as in [BP] and [ HBR ]; thus (3.36) gives an improvement in the exponent.”
The geometric sieve and the density of squarefree values of invariant polynomials (2014, background): “, the work of Heath-Brown [27] and more recently of Salberger and Wooley [37].”
Counting rational points on hypersurfaces (2004, methodology): “One might also ask about bounds of the shape NðF ;BÞ 1⁄4 OF ðB 2þeÞ, as in [9], Conjecture 1, where the implied constant is allowed to”
ON THE DISTRIBUTION OF GALOIS GROUPS (2010, background): “Note that in [2], it is assumed that F is absolutely irreducible rather than irreducible over the rationals, but as remarked in [10], top of page 556, if F is irreducible over the rationals, but not absolutely irreducible, then by Bézout’s Theorem N(F ;P1, P2) ≪d 1, so we may clearly assume that F i”
Integer points on the dilation of a subanalytic surface (2004, methodology): “Motivation for this adaptation was provided by the recent work of Heath-Brown [5], wherein a generalization of this same main tool of [3] to higher dimensions is effected by rather different means in the algebraic setting.”