Lower bounds for the rank of families of abelian varieties under base change
Volume 189 / 2019
Acta Arithmetica 189 (2019), 263-282
MSC: Primary 11G05, 11G10, 11G30, 14D10; Secondary 14H40, 14K15.
DOI: 10.4064/aa180411-4-7
Published online: 17 May 2019
Abstract
We consider the following question: given a family $\mathcal{A}$ of abelian varieties 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 $B(k)$, the rank of the fibre is at least the generic rank, i.e. the rank of $\mathcal{A}(k(B))$. When the base curve $B$ is rational, we give geometric conditions which ensure that for infinitely many fibres the rank jumps up. Examining the case of Jacobian fibrations, we show that in certain cases we get infinitely many fibres where the rank jumps by at least two units.
