Galois action on higher etale homotopy groups

To an algebraic variety over a number field F one can associate its Q_p-etale cohomology groups, equipped with an action of the absolute Galois group of F — such representations are known to enjoy several special properties that do not hold for arbitrary representations. For example, they are de Rham at p and the eigenvalues of Frobenius elements at almost all places are Weil numbers. Analogous facts hold for linear representations of the Galois group that can be extracted (e.g. by considering regular functions on the pro-algebraic completion) form the Galois action on the etale fundamental group, and one expects that all such representations arise from cohomology of algebraic varieties. In this talk, I will discuss a family of examples showing that the analogous expectation cannot hold for higher etale homotopy groups. In particular, one finds that (dual of) 2nd etale homotopy group of the moduli space of abelian varieties of dimension g>1 contains a subrepresentations that is not de Rham at p. This talk is based on joint works with Lue Pan and George Pappas.