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. We compute explicitly the normal zeta functions of the Heisenberg groups H(R), where R is a compact discrete valuation ring of characteristic zero. These zeta functions occur as Euler factors of normal zeta functions of Heisenberg groups of the form H(O K ), where O K is the ring of integers of an arbitrary number field K, at the rational primes which are non-split in K. We show that these local zeta functions satisfy functional equations upon inversion of the prime. Let F2,d denote the free class-2-nilpotent group on d generators. We compute the normal zeta functions ζ � F2,d (s), prove that they satisfy local functional equations and determine their abscissae of convergence and pole orders. Abstract We calculate the representation zeta function of the Heisenberg group over the integers of a quadratic number field. In general, the representation zeta function of a finitely generated torsion-free nilpotent group enumerates equivalence classes of representations, called twist-isoclasses. This calculation is based on an explicit description of a representative from each twist-isoclass. Our method of construction involves studying the eigenspace structure of the elements of the image of the representation and then picking a suitable basis for the underlying vector space.