Ghosts and families of abelian varieties with a common isogeny factor

Let S be an integral variety over a finitely generated field k , with generic point \eta , and A\rightarrow 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\rightarrow 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\not=p ). In particular, this enables us to show that the above local-global principle fails for certain abelian schemes built by Katz and Bültel.