Let @math be a number field with ring of integers @math . We compute explicitly the local factors of the normal zeta functions of the Heisenberg groups @math that are indexed by rational primes which are unramified in @math . We show that these local zeta functions satisfy functional equations upon the inversion of the prime.
More generally, it is of interest to compute the (normal) subgroup zeta functions of other finitely generated nilpotent groups, and their behavior under base extension. We refer the reader to @cite_6 for a comprehensive list of examples. In his MSc thesis @cite_4 , Bauer has generalized many of our results to the normal zeta functions of the higher Heisenberg groups @math for all @math , where @math is a centrally amalgamated product of @math Heisenberg groups. In other words, if @math is a ring and we view elements of @math as row vectors, and if @math denotes the @math identity matrix, then ) a , b R^m, , c R . The paper @cite_8 arose from the (uncompleted) project to compute the subgroup zeta functions @math .
@cite_4 @cite_6 @cite_8
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We study representation zeta functions of finitely generated, torsion-free nilpotent groups which are groups of rational points of unipotent group schemes over rings of integers of number fields. Using the Kirillov orbit method and @math -adic integration, we prove rationality and functional equations for almost all local factors of the Euler products of these zeta functions. We further give explicit formulae, in terms of Dedekind zeta functions, for the zeta functions of class- @math -nilpotent groups obtained from three infinite families of group schemes, generalizing the integral Heisenberg group. As an immediate corollary, we obtain precise asymptotics for the representation growth of these groups, and key analytic properties of their zeta functions, such as meromorphic continuation. We express the local factors of these zeta functions in terms of generating functions for finite Weyl groups of type @math . This allows us to establish a formula for the joint distribution of three functions, or statistics'', on such Weyl groups. Finally, we compare our explicit formulae to @math -adic integrals associated to relative invariants of three infinite families of prehomogeneous vector spaces. Let @math be a number field with ring of integers @math . We compute explicitly the local factors of the normal zeta functions of the Heisenberg groups @math that are indexed by rational primes which are unramified in @math . We show that these local zeta functions satisfy functional equations upon the inversion of the prime. We produce explicit formulae for various ideal zeta functions associated to the members of an infinite family of class- @math -nilpotent Lie rings, introduced in [8], in terms of Igusa functions. As corollaries we obtain information about analytic properties of global ideal zeta functions, local functional equations, topological, reduced, and graded ideal zeta functions, as well as representation zeta functions for the unipotent group schemes associated to the Lie rings in question.