Zeta functions and motives

Abstract

We study properties of rigid K-linear -categories A, where K is a field of characteristic 0. When A is semi-simple, we introduce a notion of multiplicities for an object of A: they are rational integers in important cases including that of pure numerical motives over a field. This yields an alternative proof of the rationality and functional equation of the zeta function of an endomorphism, and a simple proof that the number of rational points modulo q of a smooth projective variety over F q only depends on its "birational motive". The multiplicities of motives of abelian type over a finite field are equal to ±1. We also study motivic zeta functions, and an abstracted version of the Tate conjecture over finite fields.