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Point générique et saut du rang du groupe de Mordell–Weil

Volume 196 / 2020

Jean-Louis Colliot-Thélène Acta Arithmetica 196 (2020), 93-108 MSC: Primary 14K15; Secondary 14G05, 14D10. DOI: 10.4064/aa190814-18-3 Published online: 15 June 2020


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 $U$.


  • Jean-Louis Colliot-ThélèneUniversité Paris-Saclay, CNRS
    Laboratoire de Mathématiques d’Orsay
    91405 Orsay, France

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