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

2020 ◽  
Vol 71 (3) ◽  
pp. 959-980
Author(s):  
Christopher Voll
Keyword(s):  
Functional Equations ◽  
Zeta Functions ◽  
Infinite Family ◽  
Nilpotent Groups ◽  
Unipotent Group ◽  
Group Schemes ◽  
Lie Rings ◽  
Local Functional ◽  
Explicit Formulae ◽  
Local Functional Equations

Abstract 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 family of class-2 nilpotent groups, their automorphisms and pro-isomorphic zeta functions, Math. Z. 290 (2018), 909935), in terms of Igusa functions. As corollaries we obtain information about analytic properties of global ideal zeta functions, local functional equations, topological, reduced and graded ideal zeta functions, as well as representation zeta functions for the unipotent group schemes associated to the Lie rings in question.

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

2020 ◽  
Vol 30 (05) ◽  
pp. 931-975
Author(s):  
Paula Macedo Lins de Araujo
Keyword(s):  
Conjugacy Class ◽  
Functional Equations ◽  
Zeta Functions ◽  
Nilpotent Groups ◽  
Number Fields ◽  
Unipotent Group ◽  
Group Schemes ◽  
Local Factors ◽  
Joint Distributions ◽  
Rings Of Integers

This is the second of two papers introducing and investigating two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. In the first part, we proved some of their properties such as rationality and functional equations. Here, we calculate such bivariate zeta functions of three infinite families of nilpotent groups of class [Formula: see text] generalizing the Heisenberg group of ([Formula: see text])-unitriangular matrices over rings of integers of number fields. The local factors of these zeta functions are also expressed in terms of sums over finite hyperoctahedral groups, which provide formulae for joint distributions of three statistics on such groups.

scholarly journals Representation zeta functions of some nilpotent groups associated to prehomogeneous vector spaces

2017 ◽  
Vol 29 (3) ◽  
Author(s):  
Alexander Stasinski ◽  
Christopher Voll
Keyword(s):  
Zeta Functions ◽  
Nilpotent Groups ◽  
Number Fields ◽  
Vector Spaces ◽  
Unipotent Group ◽  
Finitely Generated ◽  
Group Schemes ◽  
Prehomogeneous Vector Spaces ◽  
Rings Of Integers

AbstractWe compute the representation zeta functions of some finitely generated nilpotent groups associated to unipotent group schemes over rings of integers in number fields. These group schemes are defined by Lie lattices whose presentations are modelled on certain prehomogeneous vector spaces. Our method is based on evaluating

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

2019 ◽  
Vol 22 (4) ◽  
pp. 741-774 ◽  
Author(s):  
Paula Macedo Lins de Araujo
Keyword(s):  
Functional Equations ◽  
Zeta Functions ◽  
Number Fields ◽  
Nilpotency Class ◽  
Unipotent Group ◽  
Group Schemes ◽  
Isomorphism Classes ◽  
Arithmetic Properties ◽  
Rings Of Integers ◽  
Almost All

AbstractThis 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 functions encodes the numbers of isomorphism classes of irreducible complex representations of finite dimensions of congruence quotients of the associated group and the other one encodes the numbers of conjugacy classes of each size of such quotients. In this paper, we show that these zeta functions satisfy Euler factorizations and almost all of their Euler factors are rational and satisfy functional equations. Moreover, we show that such bivariate zeta functions specialize to (univariate) class number zeta functions. In case of nilpotency class 2, bivariate representation zeta functions also specialize to (univariate) twist representation zeta functions.

scholarly journals Zeta functions of the 3-dimensional almost-Bieberbach groups

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Diego Sulca
Keyword(s):  
Zeta Function ◽  
Zeta Functions ◽  
Euler Product ◽  
3 Dimensional ◽  
Local Factors ◽  
Local Functional ◽  
Finite Sums ◽  
Complete Computation ◽  
Local Functional Equations ◽  
Torsion Free

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 factorization. We compute these series except for a finite number of local factors when the group is virtually nilpotent of Hirsch length 3. We deduce that they can be meromorphically continued to the whole complex plane and that they satisfy local functional equations. The complete computation (with no exception of local factors) is presented for those groups that are also torsion-free, that is, for the 3-dimensional almost-Bieberbach groups.

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