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.
| Original language | English |
|---|---|
| Pages (from-to) | 537-582 |
| Number of pages | 46 |
| Journal | Algebra and Number Theory |
| Volume | 18 |
| Issue number | 3 |
| DOIs | |
| State | Published - 2024 |
Bibliographical note
Publisher Copyright:© 2024, Mathematical Sciences Publishers. All rights reserved.
Funding
The research of all three authors was supported by a grant from the GIF, the German-Israeli Foundation for Scientific Research and Development (1246/2014). An extended abstract of this work for the FPSAC 2020 conference has appeared as [Carnevale et al. 2020]. Angela Carnevale gratefully acknowledges the support of the Erwin Schrödinger International Institute for Mathematics and Physics (Vienna) and the Irish Research Council through grant no. GOIPD/2018/319. The Emmy Noether Minerva Research Institute at Bar-Ilan University supported a visit by Christopher Voll during the preliminary stages of this project. Angela Carnevale and Christopher Voll are grateful to the University of Auckland for its hospitality during several phases of this project. We are grateful to Tomer Bauer for sharing with us some computations that provided important initial pointers, and to Tomer Bauer and the anonymous referee for careful readings of the text. The research of all three authors was supported by a grant from the GIF, the German-Israeli Foundation for Scientific Research and Development (1246/2014). An extended abstract of this work for the FPSAC 2020 conference has appeared as [Carnevale et al. 2020]. Angela Carnevale gratefully acknowledges the support of the Erwin Schrödinger International Institute for Mathematics and Physics (Vienna) and the Irish Research Council through grant no. GOIPD/2018/319. The Emmy Noether Minerva Research Institute at Bar-Ilan University supported a visit by Christopher Voll during the preliminary stages of this project. Angela Carnevale and Christopher Voll are grateful to the University of Auckland for its hospitality during several phases of this project.
| Funders | Funder number |
|---|---|
| Emmy Noether Research Institute for Mathematics | |
| University of Auckland | |
| German-Israeli Foundation for Scientific Research and Development | 1246/2014 |
| Irish Research Council | GOIPD/2018/319 |
Keywords
- Igusa functions
- combinatorial reciprocity theorems
- ideal growth
- ideal zeta functions
- normal zeta functions
- subgroup growth