On the multiplicities of a motive

Details On the multiplicities of a motive On the multiplicities of a motive

2006-10-14

arxiv.org/abs/math/0610446v2

part of Pure and Applied Mathematics Quarterly 5, 1 (2009) 507--570

Mathematics

Algebraic Geometry

Revised and expanded version: in particular we prove that the multiplicities of a numerical motive of abelian type over a fini

Scientific paper

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.

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