On the Bombieri-Lang Conjecture over finitely generated fields

TL;DR

This paper reduces the strong and weak Bombieri-Lang conjectures over finitely generated fields to the case where the base field is the rationals, assuming the geometric Lang conjecture.

Contribution

It establishes that the conjectures over any finitely generated field can be deduced from their cases over , assuming the geometric Lang conjecture.

Findings

Reduces strong Bombieri-Lang conjecture to  case

Reduces weak Bombieri-Lang conjecture to  case under geometric Lang conjecture

Provides a framework connecting conjectures over different fields

Abstract

The strong Bombieri-Lang conjecture postulates that, for every variety $X$ of general type over a field $k$ finitely generated over $\mathbb{Q}$, there exists an open subset $U\subset X$ such that $U(K)$ is finite for every finitely generated extension $K/k$. The weak Bombieri-Lang conjecture postulates that, for every positive dimensional variety $X$ of general type over a field $k$ finitely generated over $\mathbb{Q}$, the rational points $X(k)$ are not dense. Furthermore, Lang conjectured that every variety of general type $X$ over a field of characteristic $0$ contains an open subset $U\subset X$ such that every subvariety of $U$ is of general type, this statement is usually called geometric Lang conjecture. We reduce the strong Bombieri-Lang conjecture to the case $k=\mathbb{Q}$. Assuming the geometric Lang conjecture, we reduce the weak Bombieri-Lang conjecture to $k=\mathbb{Q}$, too.