On some questions on algebraic cycles for varieties over finite fields

Abstract: We will discuss several aspects of algebraic cycles for varieties over finite fields, which are related to the properties of their unramified cohomology groups. In particular, we will explain how to produce a variety fibred in Pfister neighbor quadrics over P^2, such that not all its codimension two Galois-invariant cycles come from the Chow group of the variety over the base field. We will also discuss some cases of the integral version of the Tate conjecture.