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On the Bombieri–Lang conjecture over finitely generated fields

### Giulio Bresciani

Vol. 16 (2022), No. 10, 2409–2414
##### Abstract

The strong Bombieri–Lang conjecture postulates that, for every variety $X$ of general type over a field $k$ finitely generated over $ℚ$, there exists an open subset $U\subset X$ such that $U\left(K\right)$ 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 $ℚ$, the rational points $X\left(k\right)$ 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 $k=ℚ$. Assuming the geometric Lang conjecture, we reduce the weak Bombieri–Lang conjecture to $k=ℚ$, too.

##### Keywords
Bombieri–Lang conjecture, varieties of general type over global fields
##### Mathematical Subject Classification
Primary: 11G35, 14G25