# Functional equations for zeta functions of groups and rings

@article{Voll2006FunctionalEF, title={Functional equations for zeta functions of groups and rings}, author={Christopher Voll}, journal={Annals of Mathematics}, year={2006}, volume={172}, pages={1181-1218} }

We introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings. The specific form of these formulae enables us to deduce local functional equations. More precisely, we prove local functional equations for the subring zeta functions associated to rings, the subgroup, conjugacy and representation zeta functions of finitely generated, torsion-free nilpotent (or T -)groups, and the normal zeta functions of T -groups of class 2. In particular we…

## 70 Citations

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.…

Stability results for local zeta functions of groups algebras, and modules

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2017

Abstract Various types of local zeta functions studied in asymptotic group theory admit two natural operations: (1) change the prime and (2) perform local base extensions. Often, the effects of both…

Stability results for local zeta functions of groups and related structures

- Mathematics
- 2015

Various types of local zeta functions studied in asymptotic group theory admit two natural operations: (1) change the prime and (2) perform local base extensions. Often, the effects of both of these…

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.

Uniformity and Functional Equations for Local Zeta Functions of Q-Split Algebraic Groups

- Mathematics
- 2008

We study the local zeta functions of an algebraic group G defined over Q together with a faithful Q-rational representation ρ. This is given by an integral over a set of p-adic points of G determined…

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…

Representation zeta functions of some compact p-adic analytic groups

- Mathematics
- 2010

Using the Kirillov orbit method, novel methods from p-adic integration and Clifford theory, we study representation zeta functions associated to compact p-adic analytic groups. In particular, we give…

Representation zeta functions of compact p-adic analytic groups and arithmetic groups

- Mathematics
- 2010

We introduce new methods from p-adic integration into the study of representation zeta functions associated to compact p-adic analytic groups and arithmetic groups. They allow us to establish that…

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…

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