Functional equations and uniformity for local zeta functions of nilpotent groups

We investigate in this paper the zeta function [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i"/] associated to a nilpotent group Γ introduced in [GSS]. This zeta function counts the subgroups H ≤ Γ whose profinite completion Ĥ is isomorphic to the profinite completion [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i"/]. By representing [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i"/] as an integral with respect to the Haar measure on the algebraic automorphism group G of the Lie algebra associated to Γ and by generalizing some recent work of Igusa [I], we give, under some assumptions on Γ, an explicit finite form for [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i"/] in terms of the combinatorial data of the root system of G and information about the weights of various representations of G. As a corollary of this finite form we are able to prove (1) a certain uniformity in p confirming a question raised in [GSS]; and (2) a functional equation that the local factors satisfy [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i"/]. This functional equation is perhaps the most important result of the paper as it is a new feature of the theory of zeta functions of groups.