Appears in collection : Journées de géométrie arithmétique de l'IHÉS

I will discuss the formulation of a variational Tate conjecture for smooth, proper families of varieties in characteristic p in terms of crystalline cycle classes, and explain the proof of the conjecture for line bundles. A key new tool is a recent continuity theorem in topological cyclic homology, which is joint with B. Dundas. I will also discuss the proof of an infinitesimal version of the conjecture, which provides an equal characteristic p analogue of the deformational p-adic Hodge conjecture of Bloch, Esnault, and Kerz.