Also appears in collection : 2019 - T2 - WS2 - Rational points on irrational varieties

I will discuss the following question, raised by Roessler and Szamuely. Let X be a a variety over a field k and A an abelian scheme over X. Assume there exists an abelian variety B over k such that for every closed point x in X, B is geometrically an isogeny factor of the fiber Ax. Then does this imply that the constant scheme B × k(η) is geometrically an isogeny factor of the generic fiber Aη ? 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.