Points of low degree on curves over function fields

TL;DR

This paper extends the classification of smooth projective curves with infinitely many points of degree up to 5 from number fields to characteristic 0 function fields, using specialization to leverage known results.

Contribution

It generalizes existing classifications from number fields to function fields of characteristic zero, employing a specialization argument.

Findings

Classification extends from number fields to function fields for degrees up to 5.

Uses specialization to reduce the problem to known cases over number fields.

Provides a unified approach for characteristic zero function fields.

Abstract

We show that the geometric classification of smooth projective curves admitting infinitely many points of degree $d\leq 5$ extends from number fields to function fields of characteristic 0. Over number fields, this classification was established by Faltings for $d=1$, Harris--Silverman for $d=2$, Abramovich--Harris for $d=3,4$ and Kadets--Vogt for $d=4,5$. Our approach uses a specialization argument to reduce the problem over function fields to the number field case.