# Uniform bound for the number of rational points on a pencil of curves

## TL;DR

This paper establishes a uniform bound on the number of rational points across a family of algebraic curves of genus at least 2, linking it to the family and Jacobian rank, using Vojta's approach.

## Contribution

It provides the first uniform bound for rational points on a family of curves of genus ≥ 2, depending only on the family and Jacobian rank, extending previous non-uniform results.

## Key findings

- Bound depends only on the family and Jacobian rank
- Uniform bounds for torsion points in the Jacobian
- Uses Vojta's approach with a height inequality

## Abstract

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 that the number of rational points is bounded only in terms of the family and the Mordell--Weil rank of the fiber's Jacobian. Our proof uses Vojta's approach to the Mordell Conjecture furnished with a height inequality due to the second- and third-named authors. In addition we obtain uniform bounds for the number of torsion in the Jacobian that lie each fiber of the family.