Ghosts and families of abelian varieties with a common isogeny factor

Abstract

Let $S$ be an integral variety over a finitely generated field $k$, with generic point $\eta$, and $A\to S$ an abelian scheme. The Hilbert irreducibility theorem and the Tate conjectures imply that the following local-global principle always holds if $k$ is infinite. Given an abelian variety $\mathfrak{A}$ over $k$, for every closed point $s\in S$, $\mathfrak{A}$ is a geometric isogeny factor of $A_s$ if and only if $\mathfrak{A}{\times }_{k}k(\eta )$ is a geometric isogeny factor of $A_{\eta }$. If $k$ is finite, the problem is more subtle. We construct an obstruction – the ghost of $A\to S$ – which completely controls the failure of the above local-global principle and is a motive built from the weight zero part of the representation of the geometric monodromy on the $\ell$-adic Tate module of $A_{\eta }$ ($\ell \ne p$). In particular, this enables us to show that the above local-global principle fails for certain abelian schemes built by Katz and Bültel.