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 $\mathcal{T}$-)groups, and the normal zeta functions of $\mathcal{T}$-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 $\mathcal{T}$-groups, respectively.