# Computing local zeta functions of groups, algebras, and modules

@article{Rossmann2016ComputingLZ, title={Computing local zeta functions of groups, algebras, and modules}, author={T. Rossmann}, journal={arXiv: Group Theory}, year={2016} }

We develop a practical method for computing local zeta functions of groups, algebras, and modules in fortunate cases. Using our method, we obtain a complete classification of generic local representation zeta functions associated with unipotent algebraic groups of dimension at most six. We also determine the generic local subalgebra zeta functions associated with $\mathfrak{gl}_2(\mathbf{Q})$. Finally, we introduce and compute examples of graded subobject zeta functions.

## 20 Citations

A Framework for Computing Zeta Functions of Groups, Algebras, and Modules

- Mathematics
- 2017

We give an overview of the author’s recent work on methods for explicitly computing various types of zeta functions associated with algebraic counting problems. Among the types of zeta functions that…

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…

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…

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…

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.

Zeta functions of the 3-dimensional almost-Bieberbach groups

- MathematicsJournal of Group Theory
- 2022

Abstract The subgroup zeta function and the normal zeta function of a finitely generated virtually nilpotent group can be expressed as finite sums of Dirichlet series admitting Euler product…

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…

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

- Mathematics
- 2019

Abstract 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…

Computing zeta functions of table algebra orders using the Bushnell-Reiner integral approach

- Mathematics
- 2022

. We investigate the generalization of Solomon’s zeta function for integral group rings to integral adjacency algebras of association schemes and to orders generated by the standard basis of an…

Zeta functions of Lie $\mathbb{F}_p$-algebras and finite $p$-groups

- Mathematics
- 2020

We introduce and study zeta functions enumerating subalgebras or ideals of rings over finite field $\mathbb{F}_p$. We demonstrate a method that explicitly computes zeta functions of Lie…

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