Generalized Igusa functions and ideal growth in nilpotent Lie rings

TL;DR

This paper introduces generalized Igusa functions to compute explicit local ideal zeta functions for a broad class of nilpotent Lie rings, unifying previous results and revealing new symmetries.

Contribution

It defines new combinatorial rational functions and applies them to derive explicit formulas for local ideal zeta functions of nilpotent Lie rings, including free class-2-nilpotent cases.

Findings

Derived explicit formulas for local ideal zeta functions

Proved self-reciprocity and functional equations for these functions

Confirmed a conjecture on the uniformity of normal zeta functions

Abstract

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.