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

We consider pro-isomorphic zeta functions of the groups $Γ(\mathcal{O}_K)$, where $Γ$ 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 $Γ$, 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.

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