Related papers: On the number of rational points on special famili…

We extend an earlier result by Dan Abramovich, showing that a conjecture of S. Lang's implies the existence of a uniform bound on the number of $K$-rational points over all smooth curves of genus $g$ defined over $K$, where $K$ is any…

The purpose of this note is to wish a happy birthday to Professor Lucia Caporaso.* We prove that Conjecture H of Caporaso et. al. ([CHarM], sec. 6) together with Lang's conjecture implies the uniformity of rational points on varieties of…

Consider a smooth, geometrically irreducible, projective curve of genus $g \ge 2$ defined over a number field of degree $d \ge 1$. It has at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that…

We prove a few uniform versions of the Mordell-Lang Conjecture and of the Shafarevich Conjecture for curves over function fields and their rational points. The main focus is on function fields having high transcendence degree over the…

We study a particular plane curve over a finite field whose normalization is of genus 0. The number of rational points of this curve achieves the Aubry-Perret bound for rational curves. The configuration of its rational points and a…

In this paper, we prove an explicit upper bound on the number of rational points on a smooth projective curve of genus at least two over a number field. This gives explicit constants in the uniform Mordell conjecture proposed by Mazur and…

In this note, we give an alternative proof of uniform boundedness of the number of integral points of smooth projective curves over a fixed number field with good reduction outside of a fixed set of primes. We use that due to…

The aim of this paper is to show how a conjectural lower bound on the canonical height function in the spirit of Lang and Silverman leads to an explicit uniform bound on the number of rational points on curves of genus $g\geq 2$ over a…

We give examples of sequences of smooth non-isotrivial curves for every genus at least two, defined over a rational function field of positive characteristic, such that the (finite) number of rational points of the curves in the sequence…

We present some results about the number of rational points on a certain family of curves defined over a finite field. In a small number of cases the curves have more rational points than expected. Fibonacci numbers make an appearance, as…

In this paper we study a family of curves obtained by fibre products of hyperelliptic curves. We then exploit this family to construct examples of curves of given genus g over a finite field Fq with many rational points. The results…

We refine a result of L. Caporaso, J. Harris and B. Mazur, and prove: Supposons que la conjecture de Lang soit vraie. Soit $K$ un corps des nombres et $g>1$ un entier. Il existe un nombre $N(K,g)$ tel que si $L$ est une extension de degr\'e…

Consider a one-parameter family of smooth, irreducible, projective curves of genus $g\ge 2$ defined over a number field. Each fiber contains at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show…

We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve…

A common practice in arithmetic geometry is that of generalizing rational points on projective varieties to integral points on quasi-projective varieties. Following this practice, we demonstrate an analogue of a result of L. Caporaso, J.…

Given an integer $\gamma\geq 2$ and an odd prime power $q$ we show that for every large genus $g$ there exists a non-singular curve $C$ defined over $\mathbb{F}_q$ of genus $g$ and gonality $\gamma$ and with exactly $\gamma(q+1)$…

For a given genus $g \geq 1$, we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus $g$ over ${\mathbb F}_q$. As a consequence of Katz-Sarnak theory, we first get for any…

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…

In this paper, we will give a uniform upper bound of the number of rational points of bounded height in non-singular curves by applying the global determinant method.

We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals. The bounds are uniform in the curve and involve the rank of the corresponding Jacobian. The method used in the proof is a…