论文标题
nilpotent群的亲晶Zeta函数，并在基本延伸下撒上环
Pro-isomorphic zeta functions of nilpotent groups and Lie rings under base extension
论文作者
论文摘要
我们考虑组$γ（\ Mathcal {o} _K）$的亲晶Zeta函数，其中$γ$是按$ \ m athbb {z} $和$ k $定义的单位组方案，在所有数字字段上都变化了。在某些条件下，我们表明这些功能具有精细的Euler分解，并由Primes $ \ Mathfrak {P} $ $ k $索引的因素，仅取决于$γ$的结构，度量$ $ [k：\ mathbb {q}] $，以及残基$ \ mathcal $ \ mathcal $} $} $ {pp}我们表明，这些因素满足了一定的统一理性，并研究了它们对$ [k：\ mathbb {q}] $的依赖性。给出了几个单一群体家庭的明确计算。其中包括一个显然是新颖的身份，涉及二十面体群体上的置换统计。
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.
