Abstract
AbstractLetXbe a smooth projective variety over a finite field$\mathbb{F}$. We discuss the unramified cohomology groupH3nr(X, ℚ/ℤ(2)). Several conjectures put together imply that this group is finite. For certain classes of threefolds,H3nr(X, ℚ/ℤ(2)) actually vanishes. It is an open question whether this holds for arbitrary threefolds. For a threefoldXequipped with a fibration onto a curveC, the generic fibre of which is a smooth projective surfaceVover the global field$\mathbb{F}$(C), the vanishing ofH3nr(X, ℚ/ℤ(2)) together with the Tate conjecture for divisors onXimplies a local-global principle of Brauer–Manin type for the Chow group of zero-cycles onV. This sheds new light on work started thirty years ago.
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