Anna CadoretFamilies of abelian varieties with a common isogeny factor
(with Akio Tamagawa).

On Friday 07–07–2017.

I will discuss the following question, raised by Roessler and Szamuely. Let S be a a variety over a field k and A an abelian scheme over S. Assume there exists an abelian variety B over k such that for every closed point s in S, B is geometrically an isogeny factor of the fiber A_s. Then does this imply that the constant scheme B\times k(\eta) is geometrically an isogeny factor of the generic fiber A_\eta? When k is not the algebraic closure of a finite field, the answer is positive and follows by standard arguments from the Tate conjectures. The interesting case is when k is finite. I will explain how, in this case, the question can be reduced to the microweight conjecture of Zarhin. This follows from a more general result, namely that specializations of motivic l-adic representation over finite fields are controlled by a `hidden motive', corresponding to the weight zero (in the sense of algebraic groups) part of the representation of the geometric monodromy.