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

TL;DR

This paper introduces two bivariate zeta functions for unipotent group schemes over number rings, analyzing their arithmetic properties, Euler factorizations, rationality, and specializations to class number and twist representation zeta functions.

Contribution

It is the first to define and study these bivariate zeta functions, establishing their Euler factorizations, rationality, and functional equations, and exploring their specializations.

Findings

Zeta functions satisfy Euler factorizations.

Most Euler factors are rational and obey functional equations.

Zeta functions specialize to class number and twist representation zeta functions.

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.