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

Mark N. Berman, Benjamin Klopsch, Uri Onn

Published 2015-11-19Version 1

Pro-isomorphic zeta functions of finitely generated nilpotent groups are analytic functions enumerating finite-index subgroups whose profinite completion is isomorphic to that of the original group. They are closely related to classical zeta functions of algebraic groups over local fields. We study pro-isomorphic zeta functions of $D^*$ groups; that is, `indecomposable' class two nilpotent groups with centre of rank two. Up to commensurability, these groups were classified by Grunewald and Segal, and can be indexed by primary polynomials whose companion matrices define commutator relations. We provide a key step towards the elucidation of the pro-isomorphic zeta functions of $D^*$ groups of even Hirsch length by describing the automorphism groups of the associated graded Lie rings. On pro-isomorphic zeta functions of $D^*$ groups of even Hirsch length Utilizing this description of the automorphism groups, we calculate the local pro-isomorphic zeta functions of groups associated to the polynomials $x^2$ and $x^3$. In both cases, the local zeta functions are uniform in the prime $p$ and satisfy functional equations. The calculation for the group associated to $x^3$ involves delicate control of solutions to polynomial equations in rings $\mathbb{Z}/p^k\mathbb{Z}$ and thus indicates an entirely new feature of pro-isomorphic zeta functions of groups.