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Title:
Non-algebraic geometrically trivial cohomology classes over finite fields
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Abstract:
The Tate conjecture is a long-standing problem in arithmetic geometry, describing algebraic cycles on an algebraic variety in terms of Galois representation on étale cohomology with coefficients in Q_l. In contrast, an integral analogue of the Tate conjecture is known to fail in general, and a natural question is whether the failure is always caused by geometry. Over a finite field, we construct the first counterexamples to this question: in codimension 2 on our examples, a geometric cycle map is surjective but an arithmetic cycle map is not. We also show positive results toward a conjecture of Colliot-Thélène and Kahn on the third unramified cohomology group for threefolds over a finite field. This is a joint work with Federico Scavia.
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