2 2 O ct 1 99 9 DEFINABLE SETS , MOTIVES AND P - ADIC INTEGRALS

Introduction 0.1. Let X be a scheme, reduced and separated, of finite type over Z. For p a prime number, one may consider the set X(Z p) of its Z p-rational points. For every n in N, there is a natural map π n : X(Z p) → X(Z/p n+1) assigning to a Z p-rational point its class modulo p n+1. The image Y n,p of X(Z p) by π n is exactly the set of Z/p n+1-rational points which can be lifted to Z p-rational points. Denote by N n,p the cardinality of the finite set Y n,p. By a result of the first author [6], the Poincaré series P p (T) := n∈N N n,p T n (0.1.1) is a rational function of T. Later Macintyre [23], Pas [25] and Denef [9] proved that the degrees of the numerator and denominator of the rational function P p (T) are bounded independently of p. One task of the present paper is to prove a much stronger uniformity result by constructing a canonical rational function P ar (T) which specializes to P p (T) for almost all p. It follows in particular from our results, that Hence a natural idea would be to try to construct P ar (T) as a series with coefficients in K 0 (Sch Q) ⊗Q, with K 0 (Sch Q) the " Grothendieck ring of algebraic varieties over Q " , defined in 1.2. However, since different varieties over a number field may have the same L-function, and we want the function P ar (T) to be canonical, we have to replace the varieties Z n,i by Chow motives and the naive Grothendieck ring K 0 (Sch Q)⊗Q by the ring K v 0 (Mot Q, ¯ Q)⊗Q, with K v 0 (Mot Q, ¯ Q) the image of K 0 (Sch Q) in the Grothendieck ring of Chow motives with coefficients in ¯ Q, as defined in 1.3. We can now state our uniformity result on the series P p (T) as follows. 0.2. Theorem. Given X as above, there exists a canonical series P ar (T) with coefficients in K v 0 (Mot Q, ¯ Q) ⊗ Q, which is a rational function of T and which specializes 1 2 JAN DENEF AND FRANÇ OIS LOESER-after taking the trace of the Frobenius on anétale realization-onto the p-adic Poincaré series P p (T), for …