Normal zeta functions of the Heisenberg groups over number rings II -- the non-split case

TL;DR

This paper explicitly computes the local normal zeta functions of Heisenberg groups over certain rings, revealing their functional equations and contributing to understanding their behavior over number fields.

Contribution

It provides explicit formulas for the normal zeta functions of Heisenberg groups over non-split primes in number fields, extending previous work to the non-split case.

Findings

Explicit formulas for local zeta functions over valuation rings.

Functional equations under prime inversion.

Application to zeta functions of Heisenberg groups over number fields.

Abstract

We compute explicitly the normal zeta functions of the Heisenberg groups $H(R)$, where $R$ is a compact discrete valuation ring of characteristic zero. These zeta functions occur as Euler factors of normal zeta functions of Heisenberg groups of the form $H(\mathcal{O}_K)$, where $\mathcal{O}_K$ is the ring of integers of an arbitrary number field~$K$, at the rational primes which are non-split in~$K$. We show that these local zeta functions satisfy functional equations upon the inversion of the prime.