Finiteness of non-constant maps over number fields

TL;DR

This paper establishes new finiteness results for non-constant morphisms between varieties over number fields, leveraging Faltings's finiteness theorems and addressing conjectures related to hyperbolicity and rational points.

Contribution

It introduces novel finiteness theorems for morphisms over number fields using moduli space techniques and Faltings's results, advancing understanding of rational points and hyperbolicity.

Findings

Proves finiteness of non-constant maps over number fields.

Connects finiteness results to Lang's conjectures.

Utilizes moduli spaces and Faltings's theorems effectively.

Abstract

Motivated by the intermediate Lang conjectures on hyperbolicity and rational points, we prove new finiteness results for non-constant morphisms from a fixed variety to a fixed variety defined over a number field by applying Faltings's finiteness results to moduli spaces of maps.