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

Authors
  • Macedo Lins de Araujo, Paula
Type
Published Article
Journal
Journal of Group Theory
Publisher
De Gruyter
Publication Date
Mar 08, 2019
Volume
22
Issue
4
Pages
741–774
Identifiers
DOI: 10.1515/jgth-2018-0115
Source
De Gruyter
License
Yellow

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 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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