Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

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 greater 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 $U$.