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.
Theorems and generalize and unify several previously known results. The most classical may be the formula for the @math -ideal zeta function of the (abelian Lie) ring @math ; cf. [Proposition 1.1] GSS 88 . The ideal zeta functions of the so-called @math were given in [Theorem 5] Voll 05 . Formulae for the ideal zeta functions of the free class- @math -nilpotent Lie rings @math on @math generators are the main result of @cite_9 . The paper @cite_19 contains formulae for all local factors of the ideal zeta functions of the Lie rings @math , i.e. the over an arbitrary number ring @math , which are indexed by primes unramified in @math . The uniform nature of these functions had already been established in [Theorem 3] GSS 88 . Formulae for factors indexed by non-split primes are given in @cite_0 . The ideal zeta functions of the Lie rings @math were computed in [Proposition 8.4] GSS 88 , whereas for the direct products @math they were computed in @cite_10 . The ideal zeta function of the Lie ring @math was computed in [Theorem 11.1] Paajanen 08 .
@cite_0 @cite_19 @cite_9 @cite_10
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We enumerate traceless square matrices over finite quotients of compact discrete valuation rings by their image sizes. We express the associated rational generating functions in terms of statistics on symmetric and hyperoctahedral groups, viz. Coxeter groups of types A and B, respectively. These rational functions may also be interpreted as local representation zeta functions associated to the members of an infinite family of finitely generated class-2-nilpotent groups. 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. 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. We develop a practical method for computing local zeta functions of groups, algebras, and modules in fortunate cases. Using our method, we obtain a complete classification of generic local representation zeta functions associated with unipotent algebraic groups of dimension at most six. We also determine the generic local subalgebra zeta functions associated with gl2(Q). Finally, we introduce and compute examples of graded subobject zeta functions.