### On the multiplicities of a motive , by Bruno Kahn

We study the multiplicities of pure motives modulo numerical equivalence,
which are defined as scalars comparing the tannakian trace with the
ring-theoretic trace. Our general set-up is that of a rigid semi-simple
tensor category such that End(1) is a field of characteristic 0. The main
result is that, due to the existence of a Weil cohomology theory (to be
defined appropriately in the general set-up), the multiplicities are
integers. This property is sufficient for the rationality (and functional
equation) of the zeta function of an (invertible) endomorphism. We also show
that the classical equivalent conditions to the Tate conjecture for pure
motives over a finite field are of category-theoretic nature in the sense
that they can be proven in the above abstract set-up.

Bruno Kahn <kahn@math.jussieu.fr >