Related papers: A uniform estimate for the density of rational poi…

We provide an asymptotic estimate for the number of rational points of bounded height on a non-singular conic over the rationals. The estimate is uniform in the coefficients of the underlying quadratic form.

We give an upper bound for the number of rational points of height at most $B$, lying on a surface defined by a quadratic form $Q$. The bound shows an explicit dependence on $Q$. It is optimal with respect to $B$, and is also optimal for…

We give uniform upper bounds for the number of rational points of height at most $B$ on non-singular complete intersections of two quadrics in $\mathbb{P}^3$ defined over $\mathbb{Q}$. To do this, we combine determinant methods with descent…

Let $X$ be an algebraic variety, defined over the rationals. This paper gives upper bounds for the number of rational points on $X$, with height at most $B$, for the case in which $X$ is a curve or a surface. In the latter case one excludes…

Let $Z$ be a projective geometrically integral algebraic variety. This paper is concerned with estimating the number of rational points on $Z$ which have height at most $B$. The bounds obtained are uniform in varieties of fixed degree and…

Let a,b,c,d be nonzero rational numbers whose product is a square, and let V be the diagonal quartic surface in PP^3 defined by ax^4+by^4+cz^4+dw^4=0. We prove that if V contains a rational point that does not lie on any of the 48 lines on…

We study the density of everywhere locally soluble diagonal quadric surfaces, parameterised by rational points that lie on a split quadric surface.

Given a projective intersection of two quadrics X in at least 9 variables, the quantitative behaviour of the rational points on X is investigated under the assumption that X contains a pair of conjugate singular points defined over the…

Let X be a variety over a number field and let f: X --> X be an "interesting" rational self-map with a fixed point q. We make some general remarks concerning the possibility of using the behaviour of f near q to produce many rational points…

Let Q be a non-singular diagonal quadratic form in at least four variables. We provide upper bounds for the number of integer solutions to the equation Q=0, which lie in a box with sides of length 2B, as B tends to infinity. The estimates…

An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski dense subset of the biprojective hypersurface $x_1y_1^2+\dots+x_4y_4^2=0$ in…

We show that the number of non-trivial rational points of height at most $B$, that lie on the cubic surface $x_1x_2x_3=x_4(x_1+x_2+x_3)^2$, has order of magnitude $B(\log B)^6$. This agrees with the Manin conjecture.

We examine the solubility of a diagonal, translation invariant, quadratic equation system in arbitrary (dense) subsets A \subset Z and show quantitative bounds on the size of A if there are no non-trivial solutions. We use the circle method…

In this paper we consider the density of rational points on the "stacky" curve $\mathcal{X}(\mathbb{P}^1;0,2;1,2;\infty,2)$ which is $\mathbb{P}^1$ with three half points, with respect to the so-called Ellenberg-Satriano-Zuerick-Brown…

Let $X \subset \mathbb{P}^n$ be a non-singular hypersurface of degree $d>1$, and let $\epsilon>0$. This paper is concerned with the conjecture that there are $O(B^{n-1+\epsilon})$ rational points on $X$ that have height at most $B$, in…

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.

In this paper, we prove a general result computing the number of rational points of bounded height on a projective variety $V$ which is covered by lines. The main technical result used to achieve this is an upper bound on the number of…

In this paper, we establish the explicit lower bound estimates for the rank of universal quadratic forms in some certain families of real cubic fields under the condition of density one. The more general results that represent all multiples…

We prove an explicit upper bound on the number of real quadratic fields that admit a universal quadratic form of a given rank, thus establishing a density zero statement. More generally, we obtain such a result for totally positive definite…

We establish the sharp estimate <<_d N^{2/d} for the number of rational points of height at most N on an irreducible projective curve of degree d. We deduce this from a result for general hypersurfaces that is sensitive to the coefficients…