# Uniform boundedness for rational points

@article{Pacelli1996UniformBF, title={Uniform boundedness for rational points}, author={Patricia L. Pacelli}, journal={Duke Mathematical Journal}, year={1996}, volume={88}, pages={77-102} }

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 number field of fixed degree $d$, and $g$ is an integer greater than 1. The bound depends only on the genus $g$ and the degree of the number field $K$.

## 34 Citations

Height Uniformity for Algebraic Points on Curves

- MathematicsCompositio Mathematica
- 2002

We recall the main result of L. Caporaso, J. Harris, and B. Mazur's 1997 paper of ‘Uniformity of rational points.’ It says that the Lang conjecture on the distribution of rational points on varieties…

Independence of rational points on twists of a given curve

- MathematicsCompositio Mathematica
- 2006

In this paper, we study bounds for the number of rational points on twists $C'$ of a fixed curve $C$ over a number field ${\mathcal K}$, under the condition that the group of ${\mathcal K}$-rational…

Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell–Weil rank

- MathematicsJournal of the European Mathematical Society
- 2018

We show that there is a bound depending only on g and [K:Q] for the number of K-rational points on a hyperelliptic curve C of genus g over a number field K such that the Mordell-Weil rank r of its…

Uniform bounds for stably integral points on elliptic curves

- Mathematics
- 1999

We show that a conjecture by Lang and Vojta regarding integral points on varieties of logarithmic general type implies the existence of a uniform bound on the number of stably S-integral points on an…

UNIFORM BOUNDS FOR STABLY INTEGRAL POINTS ON ELLIPTIC CURVES

- 1999

We show that a conjecture by Lang and Vojta regarding integral points on varieties of logarithmic general type implies the existence of a uniform bound on the number of stably S-integral points on an…

Height uniformity for integral points on elliptic curves

- Mathematics
- 2005

We recall the result of D. Abramovich and its generalization by P. Pacelli on the uniformity for stably integral points on elliptic curves. It says that the Lang-Vojta conjecture on the distribution…

Uniformity of stably integral points on principally polarized abelian varieties of dimension ≤2

- Mathematics
- 2001

The purpose of this paper is to prove, assuming that the conjecture of Lang and Vojta holds true, that there is a uniform bound on the number of stably integral points in the complement of the theta…

Stoll, Michael Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell-

- 2021

Given a curve C defined over a field K, let #C(K) denote the number of its K-rational points. By G. Faltings’ result [in: Barsotti symposium in algebraic geometry. Memorial meeting in honor of Iacopo…

Uniform boundedness of S-units in arithmetic dynamics

- Mathematics
- 2015

Let K be a number field and let S be a finite set of places of K which contains all the Archimedean places. For any �(z) 2 K(z) of degree d � 2 which is not a d-th power in K(z), Siegel's theorem…

Uniformity of stably integral points on principally polarized abelian surfaces

- Mathematics
- 1998

We prove, assuming that the conjecture of Lang and Vojta holds true, that there is a uniform bound on the number of stably integral points in the complement of the theta divisor on a principally…

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Pacelli . Uniform boundedness for rational points

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