On some differences between number fields and function fields

TL;DR

This paper explores the differences between number fields and function fields in arithmetic geometry, highlighting how isotrivial varieties and counterexamples challenge the analogy and related conjectures.

Contribution

It identifies key gaps in the analogy between number fields and function fields, especially regarding isotrivial varieties and counterexamples to conjectures.

Findings

Isotrivial varieties over function fields break the analogy.

Counterexamples to Northcott-type statements are proposed.

Explicit counterexamples to Lang and Vojta conjectures are provided in positive characteristic.

Abstract

The analogy between the arithmetic of varieties over number fields and the arithmetic of varieties over function fields is a leading theme in arithmetic geometry. This analogy is very powerful but there are some gaps. In this note we will show how the presence of isotrivial varieties over function fields (the analogous of which do not seems to exist over number fields) breaks this analogy. Some counterexamples to a statement similar to Northcott Theorem are proposed. In positive characteristic, some explicit counterexamples to statements similar to Lang and Vojta conjectures are given.