Normal Zeta Functions Research Articles

Normal zeta functions of the Heisenberg groups over number rings II — the non-split case

We compute explicitly the normal zeta functions of the Heisenberg groups H(R), where R is a compact discrete valuation ring of characteristic zero. These zeta functions occur as Euler factors of normal zeta functions of Heisenberg groups of the form H(O K ), where O K is the ring of integers of an arbitrary number field K, at the rational primes which are non-split in K. We show that these local zeta functions satisfy functional equations upon inversion of the prime.

Zeta functions and subgroup growth in $${\varvec{P}}2{\varvec{/}m}$$ P 2 / m

By means of zeta and normal zeta functions of space groups, we determine the number of subgroups, resp. normal subgroups, of the tenth crystallographic group for any given index. This enables us to draw conclusions on the subgroup growth and the degree of this group.

Zeta and normal zeta functions for a subclass of space groups

We calculate zeta and normal zeta functions of space groups with the point group isomorphic to the cyclic group of order 2. The obtained results are applied to determine the number of subgroups, resp. normal subgroups, of a given index for each of these groups.

Normal zeta functions of the Heisenberg groups over number rings I: the unramified case

Let $K$ be a number field with ring of integers $\mathcal{O}_K$. We compute explicitly the local factors of the normal zeta functions of the Heisenberg groups $H(\mathcal{O}_K)$ that are indexed by rational primes which are unramified in $K$. We show that these local zeta functions satisfy functional equations upon the inversion of the prime.

Normal Zeta Functions of Some pro-p Groups

We give an explicit formula for the number of normal subgroups of index p n of the pro-p group , and for the normal zeta function associated with this group. Let 𝒬1(s, r) be the subgroups of the Nottingham group discovered by Ershov. We observe that these groups are isospectral with . This provides us with an infinite family of noncommensurable normally isospectral groups.

Functional equations for zeta functions of groups and rings

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 equations for the subring zeta functions associated to rings, the subgroup, conjugacy and representation zeta functions of finitely generated, torsion-free nilpotent (or $\T$-)groups, and the normal zeta functions of $\T$-groups of class 2. In particular we solve the two problems posed in \cite[Section 5]{duSG/06}. We deduce our theorems from a `blueprint result' on certain $p$-adic integrals which generalises work of Denef and others on Igusa's local zeta function. The Malcev correspondence and a Kirillov-type theory developed by Howe are used to `linearise' the problems of counting subgroups and representations in $\T$-groups, respectively.

Functional equations for zeta functions of groups and rings

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 equations for the subring zeta functions associated to rings, the subgroup, conjugacy and representation zeta functions of finitely generated, torsion-free nilpotent (or T -)groups, and the normal zeta functions of T -groups of class 2. In particular we solve the two problems posed in [9, Section 5]. We deduce our theorems from a ‘blueprint result’ on certain p-adic integrals which generalises work of Denef and others on Igusa’s local zeta function. The Malcev correspondence and a Kirillov-type theory developed by Howe are used to ‘linearise’ the problems of counting subgroups and representations in T -groups, respectively.

Functional equations for local normal zeta functions of nilpotent groups

We give explicit formulae for the local normal zeta functions of torsion-free, class-2-nilpotent groups, subject to conditions on the associated Pfaffian hypersurface which are generically satisfied by groups with small centre and sufficiently large abelianization. We show how the functional equations of two types of zeta functions – the Weil zeta function associated to an algebraic variety and zeta functions of algebraic groups introduced by Igusa – match up to give a functional equation for local normal zeta functions of groups. We also give explicit formulae and derive functional equations for an infinite family of class-2-nilpotent groups known as Grenham groups, confirming conjectures of du Sautoy.

Normal subgroup growth in free class-2-nilpotent groups

Let F2, d denote the free class-2-nilpotent group on d generators. We compute the normal zeta functions prove that they satisfy local functional equations and determine their abscissae of convergence and pole orders.

Zeta functions of groups and enumeration in Bruhat-Tits buildings

We introduce a new method to calculate local normal zeta functions of finitely generated, torsion-free nilpotent groups, [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]-groups in short. It is based on an enumeration of vertices in the Bruhat-Tits building for [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]. It enables us to give explicit computations for [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]-groups of class 2 with small centres and to derive local functional equations. Examples include formulae for nonuniform normal zeta functions.

Zeta Functions of Crystallographic Groups and Analytic Continuation

We prove that the zeta function and normal zeta function of a virtually abelian group have meromorphic continuation to the whole complex plane. We do this by relating the functions to classical L-functions of arithmetic orders considered by Hey, Solomon, Bushnell and Reiner. We calculate the zeta functions and normal zeta functions of the plane crystallographic groups. As a corollary of these calculations we produce (1)examples of two isospectral residually finite groups with non-isomorphic profinite completions and even distinct lattices of subgroups; and (2)examples of non-nilpotent residually finite groups whose zeta functions enjoy an Euler product. 1991 Mathematics Subject Classification: 11M41, 20H15.