Density of rational points on a certain biprojective hypersurface

This paper is concerned with the density of rational points on the graph of a non-algebraic pfaffian function.

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.

We establish a sharp asymptotic formula for the number of rational points up to a given height and within a given distance from a hypersurface. Our main innovation is a bootstrap method that relies on the synthesis of Poisson summation, projective duality, and the method of stationary phase. This has surprising applications to counting rational points lying on the manifold; indeed, we are able to prove an analogue of Serre’s dimension growth conjecture (originally stated for projective varieties) in this general setup. As another consequence of our main counting result, we solve the generalized Baker–Schmidt problem in the simultaneous setting for hypersurfaces.

We establish sharp upper and lower bounds for the number of rational points of bounded anticanonical height on a smooth bihomogeneous threefold defined over Q and of bidegree (1, 2). These bounds are in agreement with Manin's conjecture.

We aim to count the number of rational points on cubic Ch�atelet surfaces. Our results provide evidence toward well-accepted conjectures concerning the density of rational points on such surfaces. Using rather delicate sieve arguments we give a good lower bound for the number of rational points in boxes with respect to a natural height function

Let X be an algebraic variety and let f:X−−→X be a rational self-map with a fixed point q, where everything is defined over a number field K. We make some general remarks concerning the possibility of using the behaviour of f near q to produce many rational points on X. As an application, we give a simplified proof of the potential density of rational points on the variety of lines of a cubic fourfold, originally proved by Claire Voisin and the first author in 2007.

We obtain upper bound for the density of rational points on the cyclic covers of ℙ n . As n→∞ our estimate tends to the conjectural bound of Serre.

We study a family of K3 surfaces which have a big automorphism group. We begin with generalisations of Silverman's results: construction of canonical heights, density of rational points in one orbit,… We continue the study in estimating the density of rational points on the orbiting rational curves; this estimate is compatible with Batyrev–Manin conjecture. Moreover we settle, under more geometric hypothesis, the number of rational points of such surfaces of bounded height.

Let F ∈ Z[x1, x2, x3] be an absolutely irreducible form of degree d, producing a plane curve in P. The central aim of this paper is to analyze the density of rational points on such curves, which are contained in boxes with unequal sides. We shall see below how such considerations may be used to obtain new paucity results for equal sums of two powers. Suppose that P = (P1, P2, P3) for fixed real numbers 1 ≤ P1 ≤ P2 ≤ P3, say. Then we define