Continuous cohomology classes of the group GL_n(Z_p) with coefficients in Q_p give rise to a theory of characteristic classes for étale Z_p-local systems on algebraic varieties, valued in (absolute) étale cohomology with Q_p-coefficients. These classes can be thought of as p-adic analogs of Chern–Simons characteristic classes of complex vector bundles with a flat connection.

By a theorem of Reznikov, Chern–Simons classes of all complex local systems on a smooth proper algebraic variety are torsion in degrees > 1. The same turns out to be true for p-adic characteristic classes on smooth varieties over algebraically closed fields (at least for p large enough, as compared to the rank of the local system). But for varieties over number fields and local fields these p-adic characteristic classes happen to be non-trivial even rationally, and for local systems coming from cohomology of a family of algebraic varieties they can be partially expressed in terms of the Chern classes of the corresponding Hodge bundles.

This computation of p-adic characteristic classes relies on the notion of a Chern class for pro-étale vector bundles, and on the Hodge–Tate filtration in relative p-adic Hodge theory. This is joint work with Lue Pan.