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

Abstract

Pro-isomorphic zeta functions of finitely generated nilpotent groups form one of the group-theoretic generalisations of the Riemann zeta functions. They are Dirichlet generating functions enumerating finite-index subgroups whose profinite completion is isomorphic to that of the ambient group. We study pro-isomorphic zeta functions of $D^*$-groups; these form the building blocks of finitely generated 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. 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.Comment: 29 page