Number of Points of Function Fields over Finite Fields

Definition 1. The category Mot∼ is the Karoubian envelope (or idempotent completion) of the quotient of Mot ∼ by the ideal consisting of morphisms factoring through an object of the form M ⊗L, where L is the Lefschetz motive. This is a tensor additive category. If M ∈ Mot ∼ , we denote by M̄ its image in Mot∼. Lemma 1 ([6, Lemmas 5.3 and 5.4]). Let X, Y be two smooth projective irreducible k-varieties. Then, in Motrat, we have Hom(h̄(X), h̄(Y )) = CH0(Yk(X))⊗Q. Remark 1. To any separably generated field K/k we may associate a motive h̄(K) ∈ Motorat. If K is the function field of a smooth projective varietyX, we define h̄(K) as h̄(X). In general, by de Jong’s theorem [4] we may find some finite Galois extension L/K such that L = k(X) for such an X, where the action of G = Gal(L/K) extends to X. We then