Ghosts and families of abelian varieties with a common isogeny factor
Anna Cadoret
Sorbonne University, Paris, FranceAkio Tamagawa
Kyoto University, Kyoto, Japan

Abstract
Let be an integral variety over a finitely generated field , with generic point , and an abelian scheme. The Hilbert irreducibility theorem and the Tate conjectures imply that the following local-global principle always holds if is infinite. Given an abelian variety over , for every closed point , is a geometric isogeny factor of if and only if is a geometric isogeny factor of . If is finite, the problem is more subtle. We construct an obstruction – the ghost of – 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 -adic Tate module of (). In particular, this enables us to show that the above local-global principle fails for certain abelian schemes built by Katz and Bültel.
Cite this article
Anna Cadoret, Akio Tamagawa, Ghosts and families of abelian varieties with a common isogeny factor. J. Eur. Math. Soc. (2025), published online first
DOI 10.4171/JEMS/1597