Appears in collection : Arithmetic Geometry – A Conference in Honor of Hélène Esnault on the Occasion of Her 70th Birthday

Given an étale Z_p-local system of rank n on an algebraic variety X, continuous cohomology classes of the group GL_n(Z_p) give rise to classes in (absolute) étale cohomology of the variety with coefficients in Z_p. These characteristic classes can be thought of as p-adic analogs of Chern-Simons characteristic classes of vector bundles with a flat connection. For a smooth projective variety over complex numbers, Reznikov proved that the usual Chern-Simons classes in degrees >1 of all C-local systems are torsion. It turns out that characteristic classes of étale Z_p-local systems on algebraic varieties over non-closed fields are often non-zero even rationally. In particular, if X is a smooth variety over a p-adic field, and the local system is de Rham, then its characteristic classes are related to Chern classes of the graded quotients of the Hodge filtration on the associated vector bundle with connection. This relation can be established through considering an analog of Chern classes for vector bundles on the pro-étale site of X. This is a joint work with Lue Pan.