Two coniveau filtrations and algebraic equivalence over finite fields

We extend the basic theory of the coniveau and strong coniveau filtrations to the -adic setting.By adapting the examples of Benoist-Ottem to the -adic context, we show that the two filtrations differ over any algebraically closed field of characteristic not 2. When the base field F is finite, we show that the equality of the two filtrations over the algebraic closure F has some consequences for algebraic equivalence for codimension 2 cycles over F. As an application, we prove that the third unramified cohomology group H 3 nr (X, Q /Z ) vanishes for a large class of rationally chain-connected threefolds X over F, confirming a conjecture of Colliot-Thlne and Kahn.