Generalized Igusa functions and ideal growth in nilpotent Lie rings

Angela Carnevale, Michael M. Schein, Christopher Voll

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)537-582
Number of pages46
JournalAlgebra and Number Theory
Issue number3
StatePublished - 2024

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  • Igusa functions
  • combinatorial reciprocity theorems
  • ideal growth
  • ideal zeta functions
  • normal zeta functions
  • subgroup growth


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