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

TL;DR

This paper investigates how the Mordell-Weil rank of fibers in a family of abelian varieties over a curve varies, showing that under certain conditions, infinitely many fibers have rank exceeding the generic rank.

Contribution

It extends previous work on elliptic surfaces to more general abelian varieties, establishing conditions for infinitely many fibers with higher rank than the generic rank.

Findings

Infinitely many fibers have rank greater than the generic rank under certain geometric conditions.

The results generalize known cases from elliptic surfaces to broader abelian families.

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.