Algebra Number Theory 16 (10), 2409-2414, (2022) DOI: 10.2140/ant.2022.16.2409
KEYWORDS: Bombieri–Lang conjecture, varieties of general type over global fields, 11G35, 14G25
The strong Bombieri–Lang conjecture postulates that, for every variety of general type over a field finitely generated over , there exists an open subset such that is finite for every finitely generated extension . The weak Bombieri–Lang conjecture postulates that, for every positive dimensional variety of general type over a field finitely generated over , the rational points are not dense. Furthermore, Lang conjectured that every variety of general type over a field of characteristic contains an open subset such that every subvariety of is of general type, this statement is usually called geometric Lang conjecture.
We reduce the strong Bombieri–Lang conjecture to the case . Assuming the geometric Lang conjecture, we reduce the weak Bombieri–Lang conjecture to , too.