  Let $k$ be a number field and $U$ a smooth integral $k$-variety. Let $X \to
U$ be an abelian scheme. We consider the set $\mathcal{R}$ of rational points
$m \in U(k)$ such that the Mordell-Weil rank of the fibre $U_{m}$ is strictly
bigger than the Mordell-Weil rank of the generic fibre. We prove the following
results. If the $k$-variety $X$ is $k$-unirational, then $\mathcal{R}$ is dense
for the Zariski topology on $U$. If $X$ is $k$-rational, then $\mathcal{R}$ is
not thin in $U$. This generalizes results of Billard and of Salgado. The main
idea goes back to N\'eron's thesis: use the generic point of the generic fibre
of the family.

Algebraic Geometry;Number Theory