Point g\'en\'erique et saut du rang du groupe de Mordell-Weil

Abstract

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.