Pro-isomorphic zeta functions of nilpotent groups and Lie rings under base extension

TL;DR

This paper studies pro-isomorphic zeta functions of unipotent groups over number fields, revealing their Euler decomposition, rationality properties, and explicit formulas, including a novel permutation identity.

Contribution

It establishes the Euler decomposition of these zeta functions, proves their uniform rationality, and provides explicit computations for various unipotent groups, introducing new permutation identities.

Findings

Pro-isomorphic zeta functions decompose into Euler factors indexed by primes.

Factors depend only on group structure, degree of extension, and residue field size.

Explicit formulas and a new permutation identity for hyperoctahedral groups.

Abstract

We consider pro-isomorphic zeta functions of the groups $\Gamma(\mathcal{O}_K)$, where $\Gamma$ is a unipotent group scheme defined over $\mathbb{Z}$ and $K$ varies over all number fields. Under certain conditions, we show that these functions have a fine Euler decomposition with factors indexed by primes $\mathfrak{p}$ of $K$ and depending only on the structure of $\Gamma$, the degree $[K : \mathbb{Q}]$, and the cardinality of the residue field $\mathcal{O}_K / \mathfrak{p}$. We show that the factors satisfy a certain uniform rationality and study their dependence on $[K : \mathbb{Q}]$. Explicit computations are given for several families of unipotent groups. These include an apparently novel identity involving permutation statistics on the hyperoctahedral group.