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.
The paper @cite_10 , which introduced the groups @math , computes their @math , enumerating the subgroups of finite index in @math whose profinite completions are isomorphic to that of @math . By general principles, these zeta functions also satisfy Euler product decompositions indexed by the rational primes, whose factors are rational functions in @math . In the notation of the current paper, [Theorem 1.4] BermanKlopschOnn 18 establishes that the Euler factor @math of @math at a rational prime @math , enumerating the relevant subgroups of @math of @math -power index, is of the form @math for explicitly given numerical data'' @math , for integers @math , @math , comparable to (but different from) those given in .
@cite_10
2093477771 2962768297 2802168718 2091506870
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. The pro-isomorphic zeta function ( ^ _ (s) ) of a finitely generated nilpotent group ( ) is a Dirichlet generating function that enumerates finite-index subgroups whose profinite completion is isomorphic to that of ( ). Such zeta functions can be expressed as Euler products of p-adic integrals over the ( Q _p )-points of an algebraic automorphism group associated to ( ). In this way they are closely related to classical zeta functions of algebraic groups over local fields. This is the first of two papers in which we introduce and study two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. These zeta functions encode, respectively, the numbers of isomorphism classes of irreducible complex representations of finite dimensions and the numbers of conjugacy classes of congruence quotients of the associated groups. In this paper, we show that such bivariate zeta functions specialise to (univariate) class number zeta functions. In case of nilpotency class 2, bivariate representation zeta functions also specialise to (univariate) twist representation zeta functions. Moreover, we show that these zeta functions satisfy Euler factorisations and almost all of their Euler factors satisfy rationality and functional equations. In the second part of this work, we compute the above mentioned zeta functions of three infinite families of nilpotent groups of class two and deduce some combinatorial corollaries. 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.