Related papers: Remarks about uniform boundedness of rational poin…

The paper's main result is an effective uniform bound for the finiteness statement of the Shafarevich Conjecture over function fields. Several results on the projective geometry of curves are established in the course of the proof. These…

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…

We ask questions generalizing uniform versions of conjectures of Mordell and Lang and combining them with the Morton--Silverman conjecture on preperiodic points. We prove a few results relating different versions of such questions.

Let g be an integer greater than 1. A uniform version of the Parshin-Arakelov theorem on the finiteness of the set of non-isotrivial curves of genus g over a function field, with fixed degeneracy locus, is proved. This is applied to obtain…

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…

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 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…

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,…

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 provide in this paper an upper bound for the number of rational points on a curve defined over a one variable function field over a finite field. The bound only depends on the curve and the field, but not on the Jacobian variety of the…

Using an alternative notion of good reduction, an analog of the Shafarevich theorem for elliptic curves is proved for morphisms of the projective line over number fields.

In this article we prove the explicit Mordell Conjecture for large families of curves. In addition, we introduce a method, of easy application, to compute all rational points on curves of quite general shape and increasing genus. The method…

We give an explicit uniform result on the Mordell conjecture for non-isotrivial curves over function field of characteristic 0. The proof is based on Vojta's method, and make use of Zhang's admissible adelic line bundles and a quantitative…

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…

Given a rational function of degree at least two defined over a number field k, we study the cardinality of the set of rational iterated preimages. We prove bounds for the cardinality of this set as the rational function varies in certain…

We show that there exist genus one curves of every index over the rational numbers, answering affirmatively a question of Lang and Tate. The proof is "elementary" in the sense that it does not assume the finiteness of any Shafarevich-Tate…

We prove two related concentration inequalities concerning the number of rational points of hyperelliptic curves over subsets of a finite field. In particular, we investigate the probability of a large discrepancy between the numbers of…

We prove that various arithmetic quotients of the unit ball in $\mathbb{C}^n$ are Mordellic, in the sense that they have only finitely many rational points over any finitely generated field extension of $\mathbb{Q}$. In the previously known…

In the present article we describe a class of algebraic curves on which rational functions of two arguments may reach all their possible limiting values. We also solve a similar question for functions that can be represented as a uniform…

We investigate sections of arithmetic fundamental groups of hyperbolic curves over function fields. As a consequence we prove that the anabelian section conjecture of Grothendieck holds over all finitely generated fields over $\Bbb Q$ if it…