Bounds on the number of rational points of curves in families

TL;DR

This paper provides an alternative proof for the uniform boundedness of integral points on certain algebraic curves over number fields, utilizing families of Kodaira-Parshin constructions and p-adic period maps.

Contribution

It introduces a new proof approach by assembling Kodaira-Parshin families and analyzing p-adic period maps, extending previous results on rational points.

Findings

Boundedness of integral points established

Family construction of Kodaira-Parshin used

Analysis of p-adic period maps conducted

Abstract

In this note, we give an alternative proof of uniform boundedness of the number of integral points of smooth projective curves over a fixed number field with good reduction outside of a fixed set of primes. We use that due to Bertin-Romagny, the Kodaira-Parshin families constructed by Lawrence-Venkatesh can themselves be assembled into a family. We then repeat Lawrence-Venkatesh's study of the p-adic period map, together with the comparison of nearby fibres.