Workshop on Recent Developments in Hodge Theory and O-minimality

Abstract: Given an algebraic variety over a number field F, one can attach to it its etale cohomology groups, etale fundamental group, and higher etale homotopy groups, all equipped with an action of the absolute Galois group of F. The Galois action on etale cohomology is known to satisfy several special properties: for example, it is de Rham at places above p (where we consider etale cohomology with Q_p-coefficients) and eigenvalues of Frobenius elements are Weil numbers. Analogous facts, appropriately formulated, also hold for the Galois action on the fundamental group. On the contrary, Galois action on higher etale homotopy groups turns out to fail some of these properties. In this talk, I will discuss the observation that (dual of) higher etale homotopy groups of varieties over numbers fields often contain subrepresentations that are not de Rham at p. In our examples this stems from the difference between the cohomology of an arithmetic group and its pro-finite completion, and I will also discuss this (well-known) phenomenon, which turns out to behave very differently depending on whether the ambient reductive group of the arithmetic group is of Hodge type. This talk is based on joint works with Lue Pan and George Pappas.