DEFINABLE SETS, MOTIVES AND p-ADIC INTEGRALS

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(Zp) of its Zp-rational points. For every n in N, there is a natural map πn : X(Zp)→ X(Z/p) assigning to a Zp-rational point its class modulo p. The image Yn,p of X(Zp) by πn is exactly the set of Z/p-rational points which can be lifted to Zp-rational points. Denote by Nn,p the cardinality of the finite set Yn,p. By a result of the first author [7], the Poincaré series