A nilpotent group without local functional equations for pro-isomorphic subgroups

TL;DR

This paper constructs the first example of a torsion-free finitely generated nilpotent group whose local pro-isomorphic zeta function factors do not satisfy functional equations, challenging previous expectations.

Contribution

It provides the first explicit example of a nilpotent group with local zeta factors lacking functional equations, using a novel algebraic construction via the Malcev correspondence.

Findings

The group has nilpotency class 4 and Hirsch length 25.

The local Euler factors of its zeta function do not satisfy functional equations.

This example challenges assumptions about symmetry in local zeta functions.

Abstract

The pro-isomorphic zeta function of a torsion-free finitely generated nilpotent group G enumerates finite index subgroups H such that H and G have isomorphic profinite completions. It admits an Euler product decomposition, indexed by the rational primes. We manufacture the first example of a torsion-free finitely generated nilpotent group G such that the local Euler factors of its pro-isomorphic zeta function do not satisfy functional equations. The group G has nilpotency class 4 and Hirsch length 25. It is obtained, via the Malcev correspondence, from a Z-Lie lattice L with a suitable algebraic automorphism group Aut(L).