A Variational Tate Conjecture in crystalline cohomology

Given a smooth, proper family of varieties in characteristic p > 0, and a cycle z on a fibre of the family, we consider a Variational Tate Conjecture characterising, in terms of the crystalline cycle class of z, whether z extends cohomologically to the entire family. This is a characteristic p analogue of Grothendieck's Variational Hodge Conjecture. We prove the conjecture for divisors, and an infinitesimal variant of the conjecture for cycles of higher codimension; following de Jong, the former result can be used to reduce the l-adic Tate conjecture for divisors over finite fields to the case of surfaces.