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A method of Gabber (2002) produces unramified cohomology classes in theproducts of certain varieties with an elliptic curve. The connection betweenthird unramified cohomology and integral Hodge conjecture for codimension 2cycles (2012, which builds upon results from algebraic K-Theory, then givesmany examples of such a product for which this conjecture fails. The specialcase of the product with an Enriques surface was established by Benoist andOttem (2018).

Cohomologie non ramifi\'ee dans le produit avec une courbe elliptique