PRO-ISOMORPHIC ZETA FUNCTIONS OF NILPOTENT GROUPS AND LIE RINGS UNDER BASE EXTENSION

We consider pro-isomorphic zeta functions of the groups γ(OK), where γ is a unipotent group scheme defined over Z and K varies over all number fields. Under certain conditions, we show that these functions have a fine Euler decomposition with factors indexed by primes p of K and depending only on the structure of γ, the degree [K : Q], and the cardinality of the residue field OK/p. We show that the factors satisfy a certain uniform rationality and study their dependence on [K : Q]. Explicit computations are given for several families of unipotent groups.