Related papers: Uniform bounds for rational points on hyperellipti…

We extend work of Heath-Brown and Salberger, based on the determinant method, to provide a uniform upper bound for the number of integral points of bounded height on an affine surface, which are subject to a polynomial congruence condition.…

We provide an upper bound on the uniform exponent of approximation to a triple (xi, xi^2, xi^3) by rational numbers with the same denominator, valid for any transcendental real number xi. This upper bound refines a previous result of…

We prove asymptotic formulas for the number of rational points of bounded height on certain blow-ups of the projective space.

We study the multi-height distribution of rational points of smooth, projective and split toric varieties over $\mathbf{Q}$ using the lift of the number of points to universal torsors.

We consider the problem of counting the number of rational points on the family of Kummer surfaces associated with two non-isogenous elliptic curves. For this two-parameter family we prove Manin's unity, using the presentation of the Kummer…

For a large class of isotrivial rational elliptic surfaces (with section), we show that the set of rational points is dense for the Zariski topology, by carefully studying variations of root numbers among the fibers of these surfaces. We…

We consider an elliptic surface $\pi: \mathcal{E}\rightarrow \mathbb{P}^1$ defined over a number field $k$ and study the problem of comparing the rank of the special fibres over $k$ with that of the generic fibre over $k(\mathbb{P}^1)$. We…

We study the interaction between the group law on an elliptic curve and the additive structure of $x$-coordinates of rational points on an elliptic curve. Let $E/\mathbb{Q}$ be an elliptic curve of Mordell-Weil rank $r \geq 1$, $d \geq 1$…

Let $X$ be a smooth projective variety defined over a number field $K$. We give an upper bound for the generalized greatest common divisor of a point $x\in X$ with respect to an irreducible subvariety $Y\subseteq X$ also defined over $K$.…

A rational elliptic surface with section is a smooth, rational, complex, projective surface $\mathcal{X}$ that admits a relatively minimal fibration $f: \mathcal{X}\longrightarrow \bbP^1$ such that its general fibre is a smooth irreducible…

Let $\mathbb{F}_q$ denote the finite field with $q$ elements. In this work, we use characters to give the number of rational points on suitable curves of low degree over $\mathbb{F}_q$ in terms of the number of rational points on elliptic…

We prove fibrewise versions of classical theorems of Hopf and Leray-Samelson. Our results imply the fibrewise H-triviality after rationalization of a certain class of fibrewise H-spaces. They apply, in particular, to universal adjoint…

For small odd primes $p$, we prove that most of the rational points on the modular curve $X_0(p)/w_p$ parametrize pairs of elliptic curves having infinitely many supersingular primes. This result extends the class of elliptic curves for…

Let $K$ be a field, $a, b\in K$ and $ab\neq 0$. Let us consider the polynomials $g_{1}(x)=x^n+ax+b, g_{2}(x)=x^n+ax^2+bx$, where $n$ is a fixed positive integer. In this paper we show that for each $k\geq 2$ the hypersurface given by the…

We construct families of curves which provide counterexamples for a uniform boundedness question. These families generalize those studied previously by several authors. We show, in detail, what fails in the argument of Caporaso, Harris,…

In the present article, we consider Algebraic Geometry codes on some rational surfaces. The estimate of the minimum distance is translated into a point counting problem on plane curves. This problem is solved by applying the upper bound…

We give effective bounds for the uniformity of the Iitaka fibration. These bounds follow from an effective theorem on the birationality of some adjoint linear series. In particular we derive an effective version of the main theorem in [17].

We use the Aubry-Perret bound for singular curves, a generalization of the Hasse-Weil bound, to prove the following curious result about rational functions over finite fields: Let $f(X),g(X)\in\Bbb F_q(X)\setminus\{0\}$ be such that $q$ is…

We give a sharp upper bound on the multiplicity of a fake weighted projective space with at worst canonical singularities. This is equivalent to giving a sharp upper bound on the index of the sublattice generated by the vertices of a…

We prove upper bounds on the face numbers of simplicial complexes in terms on their girths, in analogy with the Moore bound from graph theory. Our definition of girth generalizes the usual definition for graphs.