Arithmetic hyperbolicity: automorphisms and persistence

TL;DR

This paper explores the relationship between automorphism groups and rational points on projective varieties, establishing finiteness results and introducing mildly bounded varieties to study the persistence of integral points over fields.

Contribution

It proves automorphism groups are finite for varieties with torsion automorphisms and introduces mildly bounded varieties to analyze the persistence of integral points.

Findings

Automorphism groups of certain varieties are finite.

Finiteness of rational points implies finiteness of automorphisms.

Persistence of integral points over fields is characterized by mildly bounded varieties.

Abstract

We show that if the automorphism group of a projective variety is torsion, then it is finite. Motivated by Lang's conjecture on rational points of hyperbolic varieties, we use this to prove that a projective variety with only finitely many rational points has only finitely many automorphisms. Moreover, we investigate to what extent finiteness of $S$-integral points on a variety over a number field persists over finitely generated fields. To this end, we introduce the class of mildly bounded varieties and prove a general criterion for proving this persistence.