We introduce a new class of combinatorially defined rational functions and apply them to deduce explicit formulae for local ideal zeta functions associated to the members of a large class of nilpotent Lie rings which contains the free class-2-nilpotent Lie rings and is stable under direct products. Our results unify and generalize a substantial number of previous computations. We show that the new rational functions, and thus also the local zeta functions under consideration, enjoy a self-reciprocity property, expressed in terms of a functional equation upon inversion of variables. We establish a conjecture of Grunewald, Segal, and Smith on the uniformity of normal zeta functions of finitely generated free class-2-nilpotent groups.
Some of the members of the family of Lie rings @math have previously been studied with regards to related counting problems, each leading to a different class of zeta functions. We mention specifically four such classes: first, the of a (class- @math -nilpotent Lie) ring @math , enumerating the finite index subrings of @math ; second, the of @math , the finitely generated nilpotent group associated to @math via the Mal'cev correspondence, enumerating the subgroups of finite index of @math whose profinite completions are isomorphic to the one of @math ; third, the of @math , enumerating the twist-isoclasses of complex irreducible representations of @math ; fourth, the of @math , enumerating the class numbers (i.e. numbers of conjugacy classes) of congruence quotients of this group (see @cite_14 ).
@cite_14
2939052374 2093477771 2023829897 2802168718
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. 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 introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings. The specific form of these formulae enables us to deduce local functional equations. More precisely, we prove local functional equations for the subring zeta functions associated to rings, the subgroup, conjugacy and representation zeta functions of finitely generated, torsion-free nilpotent (or T -)groups, and the normal zeta functions of T -groups of class 2. In particular we solve the two problems posed in [9, Section 5]. We deduce our theorems from a ‘blueprint result’ on certain p-adic integrals which generalises work of Denef and others on Igusa’s local zeta function. The Malcev correspondence and a Kirillov-type theory developed by Howe are used to ‘linearise’ the problems of counting subgroups and representations in T -groups, respectively. 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.