Heisenberg representations [Formula: see text] of (pro-)finite groups [Formula: see text] are by definition irreducible representations of the two-step nilpotent factor group [Formula: see text] Better known are Heisenberg groups which can be understood as allowing faithful Heisenberg representations. A special feature is that [Formula: see text] will be induced by characters [Formula: see text] of subgroups in multiple ways, where the pairs [Formula: see text] can be interpreted as maximal isotropic pairs. If [Formula: see text] is a [Formula: see text]-adic number field and [Formula: see text] the absolute Galois group then maximal isotropic pairs rewrite as [Formula: see text] where [Formula: see text] is an abelian extension and [Formula: see text] a character. We will consider the extended local Artin-root-number [Formula: see text] for those [Formula: see text] which are essentially tame and express it by a formula not depending on the various maximal isotropic pairs [Formula: see text] for [Formula: see text]

中文翻译：

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海森堡表示的局部根数——一些显式结果

（亲）有限群的海森堡表示[公式：见正文] [公式：见正文]根据定义是两步幂零因子群的不可约表示 [公式：见正文] 更广为人知的是海森堡群，可以理解为允许忠实的海森堡陈述。一个特殊的特点是[公式：见文本]将由子群的字符[公式：见文本]以多种方式诱导，其中对[公式：见文本]可以解释为最大各向同性对。如果 [Formula: see text] 是 [Formula: see text]-adic number field 并且 [Formula: see text] 是绝对 Galois 群，则最大各向同性对重写为 [Formula: see text] 其中 [Formula: see text] 是阿贝尔扩展和[公式：见正文]一个字符。我们将考虑扩展的局部 Artin-root-number [公式：