Related papers: Zeta functions of virtually nilpotent groups

Let $N$ be a finitely generated nilpotent group. The subgroup zeta function $\zeta_N^{\leq}(s)$ and the normal zeta function $\zeta_N^\lhd(s)$ of $N$ are Dirichlet series enumerating the finite index subgroups or the finite index normal…

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…

In this paper we introduce some new methods to understand the analytic behaviour of the zeta function of a group. We can then combine this knowledge with suitable Tauberian theorems to deduce results about the growth of subgroups in a…

Let $G$ be a finitely generated torsion-free nilpotent group. The representation zeta function $\zeta_G(s)$ of $G$ enumerates twist isoclasses of finite-dimensional irreducible complex representations of $G$. We prove that $\zeta_G(s)$ has…

We give a sufficient criterion for generic local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms defined over number fields. This allows us, in particular, to prove various conjectures on…

The pro-isomorphic zeta function of a finitely generated nilpotent group $\Gamma$ is a Dirichlet generating function that enumerates finite-index subgroups whose profinite completion is isomorphic to that of $\Gamma$. Such zeta functions…

We study representation zeta functions of finitely generated, torsion-free nilpotent groups which are rational points of unipotent group schemes over rings of integers of number fields. Using the Kirillov orbit method and p-adic…

For a finite group $G$, we consider the zeta function $\zeta_G(s) = \sum_{H} \abs{H}^{-s}$, where $H$ runs over the subgroups of $G$. First we give simple examples of abelian $p$-group $G$ and non-abelian $p$-group $G'$ of order $p^m, \; m…

In this article we define and study a zeta function $\zeta_G$ - similar to the Hasse-Weil zeta function - which enumerates absolutely irreducible representations over finite fields of a (profinite) group $G$. The zeta function converges on…

We introduce and study multivariate zeta functions enumerating subrepresentations of integral quiver representations. For nilpotent such representations defined over number fields, we exhibit a homogeneity condition that we prove to be…

We compute the representation and class counting zeta functions for a family of torsion-free finitely generated nilpotent groups of nilpotency class $2$. These groups arise from a generalisation of one the families of unipotent groups…

Let $F_{2,d}$ denote the free class-2-nilpotent group on $d$ generators. We compute the normal zeta functions $\zeta^\triangleleft_{F_{2,d}}(s)$, prove that they satisfy local functional equations and determine their abscissae of…

In this paper we compute the subgroup zeta functions of nilpotent semi-direct products of groups of the form $$G_n=<x_1,...,x_n,y_1,...,y_{n-1}|[x_i,x_n]=y_i, 1\leq i\leq n-1, \text{all other $[,]$ trivial}>$$ and deduce local functional…

We introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings. The specific form of these formulae enables us to deduce local functional equations. More precisely, we prove local functional…

The pro-isomorphic zeta function of a torsion-free finitely generated nilpotent group G enumerates finite index subgroups H such that H and G have isomorphic profinite completions. It admits an Euler product decomposition, indexed by the…

In this paper we derive an explicit expression for the normal zeta function of class two nilpotent groups whose associated Pfaffian hypersurface is smooth. In particular, we show how the local zeta function depends on counting mod p…

We describe an effective procedure to compute the local subgroup zeta functions of the free class-$2$-nilpotent groups on $d$ generators, for all $d$. For $d=4$, this yields a new, explicit formula. For $d\in\{4,5\}$, we compute the…

Let $\Gamma$ be a group and $r_n(\Gamma)$ the number of its $n$-dimensional irreducible complex representations. We define and study the associated representation zeta function $\calz_\Gamma(s) = \suml^\infty_{n=1} r_n(\Gamma)n^{-s}$. When…

The pro-isomorphic zeta function of a finitely generated nilpotent group is a Dirichlet generating series that enumerates all finite-index subgroups whose profinite completion is isomorphic to that of the ambient group. We study the…

We introduce a new class of combinatorially defined rational functions and apply them to deduce explicit formulae for local ideal zeta functions associated to the members of a large class of nilpotent Lie rings which contains the free…