On some differences between number fields and function fields. (English) Zbl 1378.14028

Gasbarri, Carlo

Riv. Mat. Univ. Parma (N.S.) 7, No. 1, 1-18 (2016).

Summary: 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 does not seem 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.

MSC:

14G40	Arithmetic varieties and schemes; Arakelov theory; heights
14G22	Rigid analytic geometry
11G50	Heights
14H05	Algebraic functions and function fields in algebraic geometry

Keywords:

arithmetic over function fields; height theory; Lang and Vojta conjectures