We give a sufficient criterion for generic local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms defined over number fields. This allows us, in particular, to prove various conjectures on such functional equations for ideal zeta functions of nilpotent Lie lattices. Via the Mal'cev correspondence, these results have corollaries pertaining to zeta functions enumerating normal subgroups of finite index in finitely generated nilpotent groups, most notably finitely generated free nilpotent groups of any given class.
Our proof of Theorem , presented in , proceeds by adapting the @math -adic machinery developed in @cite_19 . There, this technique is applied to establish generic local functional equations for a range of zeta functions of groups and rings. The most general of these applications is to subring zeta functions of arbitrary rings of finite additive rank, i.e. finitely generated abelian groups with some bi-additive multiplicative structure; see [Theorem A] Voll 10 . Via the Mal'cev correspondence, this translates into results for the generic Euler factors of the subgroup zeta functions of finitely generated nilpotent groups, i.e. Dirichlet generating series enumerating all finite index subgroups of such a group; see [Corollary 1.1] Voll 10 . In [Theorem C] Voll 10 we prove functional equations for generic local ideal zeta functions of nilpotent Lie rings of class @math (or, equivalently, generic local normal zeta functions of finitely generated nilpotent groups of class @math ). Theorem generalizes this result.
@cite_19
2023829897
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.