# Families of abelian varieties with a common isogeny factor

## Anna Cadoret

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.