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

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.

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

International Journal of Algebra and Computation ◽

10.1142/s0218196720500265 ◽

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.

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

Forum Mathematicum ◽

10.1515/forum-2015-0099 ◽

2017 ◽

Vol 29 (3) ◽

Cited By ~ 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

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

The Quarterly Journal of Mathematics ◽

10.1093/qmathj/haaa010 ◽

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.

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

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004117000585 ◽

2017 ◽

Vol 165 (3) ◽

pp. 435-444 ◽

Cited By ~ 2

Author(s):

TOBIAS ROSSMANN

Keyword(s):

Group Theory ◽

Functional Equations ◽

Zeta Functions ◽

Local Zeta Functions ◽

Single Formula ◽

Almost All ◽

Stability Results ◽

Zeta Functions Of Groups ◽

Local Base

AbstractVarious 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 the preceding operations can be expressed simultaneously in terms of a single formula, a statement made precise using what we call local maps of Denef type. We show that assuming the existence of such formulae, the behaviour of local zeta functions under variation of the prime in a set of density 1 in fact completely determines these functions for almost all primes and, moreover, it also determines their behaviour under local base extensions. We discuss applications to topological zeta functions, functional equations, and questions of uniformity.

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