MOTIVES OVER FINITE FIELDS

The category of motives over the algebraic closure of a finite field is known to be a semisimple Q-linear Tannakian category, but unless one assumes the Tate conjecture there is little further one can say about it. However, once this conjecture is assumed, it is possible to give an almost entirely satisfactory description of the category together with its standard fibre functors. In particular it is possible to list properties of the category that characterize it up to equivalence and to prove (without assuming any conjectures) that there does exist a category with these properties. The Hodge conjecture implies that there is a functor from the category of CM-motives over Qal to the category of motives over F. We construct such a functor.

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