On some differences between number fields and function fields

2016/06/21 by Carlo Gasbarri, Gasbarri, Carlo
Mathematics · #11G50 #14G22 #14G40 #Algebraic Geometry (math.AG) #Analytic Number Theory Research #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1606.06517

openalex publication_date 2016/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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