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

## TL;DR

This paper studies the distribution of points in a family of abelian varieties over a number field where the Mordell-Weil rank jumps, showing density or non-thinness depending on the rationality properties of the total space.

## Contribution

It generalizes previous results by establishing the density and non-thinness of points with increased Mordell-Weil rank based on the rationality of the total space.

## Key findings

- If the total space is $k$-unirational, the set of points with rank jump is Zariski dense.
- If the total space is $k$-rational, this set is not thin in $U$.
- The approach uses the generic point of the generic fibre, inspired by Néron's thesis.

## 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.