On pro-isomorphic zeta functions of $D^*$-groups of even Hirsch length

TL;DR

This paper analyzes pro-isomorphic zeta functions of specific nilpotent groups, providing explicit descriptions, functional equations, and conjectures about their symmetries, with implications for understanding their global properties.

Contribution

It offers a complete description of local pro-isomorphic zeta functions for certain $D^*$-groups, introduces new functional equations, and formulates a conjecture on symmetry factors in these functions.

Findings

Local zeta functions are uniform in prime p for groups associated to t^2 and t^3.

Functional equations are satisfied, revealing symmetries not predicted by existing theory.

Results extend to infinite families of class-two nilpotent groups.

Abstract

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 pro-isomorphic zeta functions of $\mathbb{Q}$-indecomposable $D^*$-groups of even Hirsch length. These groups are building blocks of finitely generated class-two nilpotent groups with rank-two centre, up to commensurability. Due to a classification by Grunewald and Segal, they are parameterised by primary polynomials whose companion matrices define commutator relations for an explicit presentation. For Grunewald-Segal representatives of even Hirsch length of type $f(t)=t^m$, we give a complete description of the algebraic automorphism groups of associated Lie lattices. Utilising the automorphism groups, we determine the local pro-isomorphic zeta functions of groups associated to $t^2$ and $t^3$. In both cases, the local zeta functions are uniform in the prime $p$ and satisfy functional equations. The functional equations for these groups, not predicted by the currently available theory, prompt us to formulate a conjecture which prescribes, in particular, information about the symmetry factor appearing in local functional equations for pro-isomorphic zeta functions of nilpotent groups. Our description of the local zeta functions also yields information about the analytic properties of the corresponding global pro-isomorphic zeta functions. Some of our results for the $D^*$-groups associated to $t^2$ and $t^3$ generalise to two infinite families of class-two nilpotent groups that result naturally from the initial groups via `base extensions'.