On the multiplicities of a motive
  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 (IMJ)