Pro-isomorphic zeta functions of some $D^\ast$ Lie lattices of even rank

TL;DR

This paper computes local pro-isomorphic zeta functions for a family of class-two-nilpotent Lie lattices of even rank, revealing a pattern of rational functions with functional equations, extending understanding of zeta functions in algebraic structures.

Contribution

It provides explicit formulas for pro-isomorphic zeta functions of certain Lie lattices, connecting combinatorial rational functions with algebraic properties, and generalizes previous results to a broader family.

Findings

Explicit formulas for zeta functions at almost all primes.

Rational functions satisfy a functional equation under variable inversion.

Connects combinatorial structures with algebraic zeta functions.

Abstract

We compute the local pro-isomorphic zeta functions at all but finitely many primes for a certain family of class-two-nilpotent Lie lattices of even rank, parametrized by irreducible non-linear polynomials $f(x) \in \mathbb{Z} [x]$, that corresponds to a family of groups introduced by Grunewald and Segal. The result is expressed in terms of a combinatorially defined family of rational functions satisfying a functional equation upon inversion of the variables.