Functional equations for zeta functions of groups and rings

Abstract

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 --)groups, and the normal zeta functions of --groups of class 2. 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 --groups, respectively.

Zeta functions of three-dimensional p-adic Lie algebras

“…We point out that Theorem 1.1 is a 'local result', whereas the main conclusions of the results in [20] are valid for almost all completions of a 'global object' (such as a torsion-free nilpotent group or a torsion-free ring). The main application of the approach developed in [20] is to prove that, given a torsion-free ring L (not necessarily associative or Lie), the associated local zeta function ζ Z p ⊗ Z L (s) satisfies a functional equation for almost all primes p. The occurrence of functional equations for the zeta functions of some three-dimensional Z p -Lie algebras is therefore only explained by the results of [20] in case these algebras are the 'generic' completions of an algebra over Z (such as, for example, sl 2 (Z p ) for odd p). This corresponds to the fact that the proof of a functional equation for Igusa's local zeta function given in [2] critically depends on good reduction modulo p. 4.…”

“…In [20], Voll introduced a method for computing zeta functions of Z p -algebras in terms of certain p-adic integrals generalising Igusa's local zeta function. Given a polynomial f (x) ∈ Z p [x 1 , .…”

Reduced zeta functions of Lie algebras

“…Recently, C. Voll [8] has proved that a functional equation exists for all ideal zeta functions of Lie algebras of nilpotency class at most 2 and for all subalgebra zeta functions. Let L be a Lie algebra over Z such that…”

A nilpotent group without local functional equations for pro-isomorphic subgroups

“…One of the prominent questions driving the subject has been whether the local zeta functions satisfy functional equations: is it true that ζ * Γ,p (s)| p→p −1 = (−1) a p b−cs ζ Γ,p (s) for almost all p, where a, b, c are certain integer parameters depending on Γ? In [10], Voll derived positive answers for subgroup zeta functions in general and for normal zeta functions associated to groups of nilpotency class at most 2. More precisely, he showed that the local zeta functions in question can be expressed as rational functions in p −s whose coefficients involve the numbers b V (p) of F p -rational points of certain smooth projective varieties V .…”

“…One of the key results in [5] is that each of the local factors defined in (1.1) is in fact a rational function over Q in p −s . Over the last 20 years many important advances have been made in the study of zeta functions of nilpotent groups; for instance, see [2,4,10]. One of the prominent questions driving the subject has been whether the local zeta functions satisfy functional equations: is it true that ζ * Γ,p (s)| p→p −1 = (−1) a p b−cs ζ Γ,p (s) for almost all p, where a, b, c are certain integer parameters depending on Γ?…”