Download this article
 Download this article For screen
For printing
Recent Issues

Volume 18
Issue 5, 847–1038
Issue 4, 631–846
Issue 3, 409–629
Issue 2, 209–408
Issue 1, 1–208

Volume 17, 12 issues

Volume 16, 10 issues

Volume 15, 10 issues

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
Other MSP Journals
Generalized Igusa functions and ideal growth in nilpotent Lie rings

Angela Carnevale, Michael M. Schein and Christopher Voll

Vol. 18 (2024), No. 3, 537–582

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.

subgroup growth, ideal growth, normal zeta functions, ideal zeta functions, Igusa functions, combinatorial reciprocity theorems
Mathematical Subject Classification
Primary: 05A15, 11M41, 20E07
Received: 13 June 2022
Revised: 28 February 2023
Accepted: 13 April 2023
Published: 16 February 2024
Angela Carnevale
School of Mathematical and Statistical Sciences
University of Galway
Michael M. Schein
Department of Mathematics
Bar Ilan University
Ramat Gan
Christopher Voll
Faculty of Mathematics
Bielefeld University

Open Access made possible by participating institutions via Subscribe to Open.