[Paper Review] Analytic properties of bivariate representation and conjugacy class zeta functions of finitely generated nilpotent groups

Paula Macedo Lins de Araujo|arXiv (Cornell University)|Jul 15, 2018

Advanced Algebra and Geometry13 references4 citations

TL;DR

This paper establishes that the domains of convergence and meromorphic continuation of bivariate representation and conjugacy class zeta functions for finitely generated nilpotent groups derived from unipotent group schemes over rings of integers of number fields are independent of the underlying number field, up to finitely many local factors. The result relies on analyzing local zeta functions at prime ideals and proving uniform convergence and meromorphy across all number fields beyond a finite exceptional set.

ABSTRACT

Let $\mathbf{G}$ be a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring $\mathcal{O}$ of integers of a number field. We consider bivariate zeta functions of groups of the form $\mathbf{G}(\mathcal{O})$ encoding, respectively, the numbers of isomorphism classes of irreducible complex representations of finite dimensions and the numbers of conjugacy classes of congruence quotients of the associated groups. We show that the domains of convergence and meromorphic continuation of these zeta functions of groups $\mathbf{G}(\mathcal{O})$ are independent of the number field $\mathcal{O}$ considered, up to finitely many local factors.

Motivation & Objective

• To study the analytic behavior of bivariate zeta functions encoding representation and conjugacy class data for finitely generated nilpotent groups.
• To determine whether the domains of convergence and meromorphic continuation of these zeta functions depend on the choice of number field.
• To show that such analytic properties are independent of the number field, up to finitely many local factors.
• To establish uniformity in the convergence and meromorphy of zeta functions across different rings of integers of number fields.
• To analyze the local zeta functions at prime ideals and their global product structure.

Proposed method

• Define bivariate zeta functions for groups $\mathbf{G}(\mathcal{O})$ using congruence quotients and dimensions/conjugacy class sizes.
• Express the global zeta functions as Euler products over prime ideals $\mathfrak{p}$ of the ring of integers $\mathcal{O}$.
• Analyze the local zeta functions $\mathcal{Z}^{\ast}_{\mathbf{G}(\mathcal{O}_{\mathfrak{p}})}(s_1,s_2)$ for each prime $\mathfrak{p}$, using convergence conditions derived from representation-theoretic and group-theoretic data.
• Prove that the domains of convergence and meromorphic continuation of the global zeta functions are independent of $\mathcal{O}$, up to finitely many local factors.
• Use domain decomposition and intersection arguments to show that the convergence domain contains a uniform neighborhood of the critical strip, independent of the number field.
• Leverage the structure of $\mathscr{D}_{\mathscr{R},\delta}$ domains to establish uniform convergence and meromorphy beyond a finite set of primes.

Experimental results

Research questions

• RQ1 Does the domain of convergence of the bivariate representation zeta function $\mathcal{Z}^{\textup{irr}}_{\mathbf{G}(\mathcal{O})}(s_1,s_2)$ depend on the number field $K$?
• RQ2 Is the meromorphic continuation of the bivariate conjugacy class zeta function $\mathcal{Z}^{\textup{cc}}_{\mathbf{G}(\mathcal{O})}(s_1,s_2)$ independent of the ring of integers $\mathcal{O}$ of a number field?
• RQ3 Can the analytic behavior of these zeta functions be made uniform across all number fields, up to finitely many local factors?
• RQ4 What is the precise structure of the domain of convergence for the global zeta functions in terms of local convergence conditions?
• RQ5 How do the local zeta functions at prime ideals contribute to the global analytic structure of the zeta functions?

Key findings

• The domains of convergence and meromorphic continuation of the bivariate zeta functions $\mathcal{Z}^{\textup{irr}}_{\mathbf{G}(\mathcal{O})}(s_1,s_2)$ and $\mathcal{Z}^{\textup{cc}}_{\mathbf{G}(\mathcal{O})}(s_1,s_2)$ are independent of the number field $K$, up to finitely many local factors.
• There exists a finite set $\mathcal{Q}$ of prime ideals such that the global zeta functions $\mathcal{Z}^{\ast}_{\mathbf{G}(\mathcal{O})}^{\mathcal{Q}}(s_1,s_2)$ have domains of convergence and meromorphic continuation that do not depend on $\mathcal{O}$.
• The convergence domain $\mathscr{D}_{\mathscr{R},\delta}$ is uniform across all number fields, with $\delta > 0$ independent of $\mathcal{O}$, ensuring analytic uniformity.
• The global zeta functions are meromorphic on a domain $\mathscr{M}_{\mathbf{G}}^{\ast}$ that strictly contains the critical domain $\mathscr{D}_{\mathbf{G}}^{\ast}$, independent of $\mathcal{O}$.
• The local zeta functions $\mathcal{Z}^{\ast}_{\mathbf{G}(\mathcal{O}_{\mathfrak{p}})}(s_1,s_2)$ are uniformly convergent and meromorphic in a common domain for all $\mathfrak{p}$ outside a finite set $\mathcal{Q}$.
• The product structure $\prod_{\mathfrak{p} \notin \mathcal{Q}} \mathcal{Z}^{\ast}_{\mathbf{G}(\mathcal{O}_{\mathfrak{p}})}(s_1,s_2)$ converges and extends meromorphically to a domain independent of $\mathcal{O}$.