Unboundedness of the number of rational points on curves over function fields

TL;DR

This paper demonstrates that for certain sequences of algebraic curves over function fields in positive characteristic, the number of rational points can grow without bound, challenging previous assumptions of uniform boundedness.

Contribution

It provides explicit examples of non-isotrivial curves over function fields with unbounded rational points, highlighting new phenomena in arithmetic geometry.

Findings

Sequences of non-isotrivial curves with unbounded rational points

Counterexamples to uniform boundedness conjectures

Applicable for all genera at least two

Abstract

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 cannot be uniformly bounded.