## Jean-Louis Colliot-Thélène, CNRS and Paris-Saclay University (Orsay)

December 11th, 2020 2:30-3:30 pm CET

**Title** : Jumps in the rank of the Mordell-Weil group

**Abstract** : Let $k$ be a number field and $U$ a smooth integral $k$-variety.
Let $X$→$U$ be an abelian scheme. We consider the set $U(k{)}_{+}$⊂$U(k)$ of $k$-rational points of $U$ such that the Mordell-Weil rank of the fibre ${X}_{m}$ is strictly bigger than the Mordell-Weil rank of the generic fibre over the function field $k(U)$.

We prove : if the $k$-variety $X$ is $k$-unirational, then $U(k{)}_{+}$ is dense for the Zariski topology on $U$. Variants are given and compared with old and new results in the literature.