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# Families of abelian varieties with a common isogeny factor

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 $$A_x$$. 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.