# Generalized Igusa functions and ideal growth in nilpotent Lie rings

@article{Carnevale2019GeneralizedIF, title={Generalized Igusa functions and ideal growth in nilpotent Lie rings}, author={Angela Carnevale and Michael M. Schein and Christopher Voll}, journal={arXiv: Rings and Algebras}, year={2019} }

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…

## Figures from this paper

## 7 Citations

IDEAL ZETA FUNCTIONS ASSOCIATED TO A FAMILY OF CLASS-2-NILPOTENT LIE RINGS

- MathematicsThe Quarterly Journal of Mathematics
- 2020

We produce explicit formulae for various ideal zeta functions associated to the members of an infinite family of class-$2$-nilpotent Lie rings, introduced in M. N. Berman, B. Klopsch and U. Onn (A…

Ideal growth in amalgamated powers of nilpotent rings and zeta functions of quiver representations

- Mathematics
- 2022

Let L be a nilpotent algebra of class two over a compact discrete valuation ring A of characteristic zero or of sufficiently large positive characteristic. Let q be the residue cardinality of A. The…

Pro-isomorphic zeta functions of nilpotent groups and Lie rings under base extension

- Mathematics
- 2020

We consider pro-isomorphic zeta functions of the groups $\Gamma(\mathcal{O}_K)$, where $\Gamma$ is a nilpotent group scheme defined over $\mathbb{Z}$ and $K$ varies over all number fields. Under…

Zeta functions of integral nilpotent quiver representations

- Mathematics
- 2020

We introduce and study zeta functions enumerating subrepresentations of integral quiver representations. For nilpotent such representations defined over number fields, we exhibit a homogeneity…

A remark on the degree of polynomial subgroup growth of nilpotent groups

- Mathematics
- 2021

We show that if two finitely generated nilpotent groups have isomorphic C-Mal’cev completions, then their subgroup (resp. normal) zeta functions have the same abscissa of convergence. A similar…

The cotype zeta function of $\mathbb{Z}^d$

- Mathematics
- 2017

We give an asymptotic formula for the number of sublattices $\Lambda \subseteq \mathbb{Z}^d$ of index at most $X$ for which $\mathbb{Z}^d/\Lambda$ has rank at most $m$, answering a question of Nguyen…

Enumerating conjugacy classes of graphical groups over finite fields

- MathematicsBulletin of the London Mathematical Society
- 2022

Throughout, graphs are finite, simple, and (unless otherwise indicated) contain at least one vertex. When the reference to an ambient graph is clear, we use ∼ to indicate the associated adjacency…

## References

SHOWING 1-10 OF 53 REFERENCES

IDEAL ZETA FUNCTIONS ASSOCIATED TO A FAMILY OF CLASS-2-NILPOTENT LIE RINGS

- MathematicsThe Quarterly Journal of Mathematics
- 2020

We produce explicit formulae for various ideal zeta functions associated to the members of an infinite family of class-$2$-nilpotent Lie rings, introduced in M. N. Berman, B. Klopsch and U. Onn (A…

Functional equations for local normal zeta functions of nilpotent groups

- Mathematics
- 2003

Abstract.We give explicit formulae for the local normal zeta functions of torsion-free, class-2-nilpotent groups, subject to conditions on the associated Pfaffian hypersurface which are generically…

Geometric structure of class two nilpotent groups and subgroup growth

- Mathematics
- 2008

In this paper we derive an explicit expression for the normal zeta function of class two nilpotent groups whose associated Pfaffian hypersurface is smooth. In particular, we show how the local zeta…

Local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms

- Mathematics
- 2016

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…

Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type B

- Mathematics
- 2011

We study representation zeta functions of finitely generated, torsion-free nilpotent groups which are groups of rational points of unipotent group schemes over rings of integers of number fields.…

Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, II: Groups of type F, G, and H

- MathematicsInt. J. Algebra Comput.
- 2020

Here, such bivariate zeta functions of three infinite families of nilpotent groups of class 2 generalising the Heisenberg group of three by three unitriangular matrices over rings of integers of number fields are calculated.

Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, I: Arithmetic properties

- MathematicsJournal of Group Theory
- 2019

This is the first of two papers in which we introduce and study two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields.
One of these zeta…

Normal zeta functions of the Heisenberg groups over number rings II — the non-split case

- Mathematics
- 2014

We compute explicitly the normal zeta functions of the Heisenberg groups H(R), where R is a compact discrete valuation ring of characteristic zero. These zeta functions occur as Euler factors of…

Representation growth of some torsion-free finitely generated $2$-nilpotent groups

- Mathematics
- 2017

We devise a method for computing representation zeta functions of torsion-free finitely generated $2$-nilpotent groups whose associated Lie lattices have an extra smoothness condition. This method is…

Igusa-type functions associated to finite formed spaces and their functional equations

- Mathematics
- 2006

We study symmetries enjoyed by the polynomials enumerating non-degenerate flags in finite vector spaces, equipped with a non-degenerate alternating bilinear, hermitian or quadratic form. To this end…