ZETA-FUNCTIONS OF GROUPS AND RINGS - UNIFORMITY

There are various natural local zeta functions associated with groups and rings for each prime p. We consider the question of how these functions behave as we vary the prime p and the groups (or rings) range over a specific class of groups (or rings), e.g. finitely generated torsion-free nilpotent groups of a fixed Hirsch length or p-adic analytic groups of a fixed dimension. Using a result of Macintyre's on the uniformity of parameterized p-adic integrals, together with various natural parameter spaces we define for these classes of groups, we prove a strong finiteness theorem on the possible poles of these local zeta functions.