On the number of rational points on special families of curves over function fields

TL;DR

This paper constructs specific families of algebraic curves over function fields that serve as counterexamples to uniform boundedness, analyzing why certain conjectures fail and providing explicit bounds on rational points.

Contribution

It introduces new families of curves over function fields, demonstrating counterexamples to uniform boundedness and analyzing the failure of existing arguments.

Findings

Counterexamples to uniform boundedness over function fields

Explicit bounds for heights and rational points

Analysis of the failure in Caporaso-Harris-Mazur argument

Abstract

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, Mazur that uniform boundedness follows from the Lang conjecture. We also give a direct proof that these curves have finitely many rational points and give explicit bounds for the heights and number of such points.