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

Berman, Mark; Glazer, Itay; Schein, Michael

doi:10.1090/tran/8506

Pro-isomorphic zeta functions of nilpotent groups and Lie rings under base extension
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by Mark N. Berman, Itay Glazer and Michael M. Schein PDF

Trans. Amer. Math. Soc. 375 (2022), 1051-1100 Request permission

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.

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