  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.
