Classes de cycles motiviques \'etales

TL;DR

This paper investigates the structure of unramified étale cohomology groups of smooth varieties over various fields, revealing their composition and conditions under which certain parts vanish, with implications for the Tate conjecture.

Contribution

It provides a detailed description of unramified étale H^3 groups over different fields, linking them to cycle maps, Griffiths groups, and verifying cases of the Tate conjecture.

Findings

For separably closed, finite, or p-adic fields, H^3 unramified étale cohomology is an extension of a finite group by a divisible group.

When the base field is finite and the variety is of abelian type, the divisible part D vanishes, confirming the Tate conjecture in this case.

In certain cases over finite fields, the divisible part D is zero if H^3 is of coniveau > 0, but this is not always necessary.

Abstract

Let X be a smooth variety over a field k, and l be a prime number invertible in k. We study the (\'etale) unramified H^3 of X with coefficients Q_l/Z_l(2) in the style of Colliot-Th\'el\`ene and Voisin. If k is separably closed, finite or p-adic, this describes it as an extension of a finite group F by a divisible group D, where F is the torsion subgroup of the cokernel of the l-adic cycle map. If k is finite and X is projective and of abelian type, verifying the Tate conjecture, D=0. If k is separably closed, we relate D to an l-adic Griffiths group. If k is the separable closure of a finite field and X comes from a variety over a finite field as described above, then D = 0 as soon as H^3(X,Q_l) is entirely of coniveau > 0, but an example of Schoen shows that this condition is not necessary.