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

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.

24 pages. Preprint of an article published in the Journal of Group Theory

References in corpus (5)

• The average size of the kernel of a matrix and orbits of linear groups
• Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, II: Groups of type F, G, and H
• The average size of the kernel of a matrix and orbits of linear groups, II: duality
• Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, I: Arithmetic properties
• Analytic properties of bivariate representation and conjugacy class zeta functions of finitely generated nilpotent groups

Cited by in corpus (5)

• The average size of the kernel of a matrix and orbits of linear groups
• Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, II: Groups of type F, G, and H
• The average size of the kernel of a matrix and orbits of linear groups, II: duality
• Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, I: Arithmetic properties
• Univariate and bivariate zeta functions of unipotent group schemes of type $G$