Équivalence rationnelle, équivalence numérique et produits de courbes elliptiques sur un corps fini, by Bruno Kahn

We prove that if X is a product of elliptic curves over a finite field k, rational and numerical equivalences agree on X. The proof uses an idea of Soule idea, U. Jannsen's semi-simplicity theorem for numerical motives, M. Spiess's proof of the Tate conjecture for such varieties and a result of Y. Andre and the author inspired by S.I. Kimura's results on finite-dimensional Chow motives. We give some consequences, among which: the conjectures of Lichtenbaum on special values of zeta functions hold true for X, the second Chow group of X is finitely generated, the Beilinson-Soule conjecture holds in weight n for the function field of X provided n or dim X is at most 2. Also, if U is an open subset of X with dim X at most 2, the action of Frobenius on the l-adic cohomology of U with compact supports is semi-simple and its characteristic polynomial does not depend on the prime l (different from the characteristic of k).

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