Lower bounds for the rank of families of abelian varieties under base change

Marc Hindry; Cecília Salgado

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arXiv · 1712.05858

Lower bounds for the rank of families of abelian varieties under base change

Marc Hindry·Cecília Salgado

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Abstract

We consider the following question : given a family over abelian varieties $\mathcal{A}$ over a curve $B$ defined over a number field $k$, how does the rank of the Mordell-Weil group of the fibres $\mathcal{A}_t(k)$ vary? A specialisation theorem of Silverman guarantees that, for almost all $t$ in $C(k)$, the rank of the fibre is at least the generic rank, that is the rank of $\mathcal{A}(k(B))$. When the base curve $B$ is rational, we show, at least in many cases and under some geometric conditions, that there are infinitely many fibres for which the rank is larger than the generic rank. This paper is a sequel to a paper of the second author where the case of elliptic surfaces is treated.

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Marc Hindry, Cecília Salgado. 2017-12-15. Lower bounds for the rank of families of abelian varieties under base change. https://arxiv.org/abs/1712.05858

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