We compute all the moments of a normalization of the function that counts unramified |$H_{8}$|-extensions of quadratic fields, where |$H_{8}$| is the quaternion group of order |$8$|⁠, and show that the values of this function determine a point mass distribution. As a consequence toward non-abelian Cohen–Lenstra heuristics of 2-groups, we show this implies that the probability is 0 for any fixed group to appear as the subgroup of the Galois group of the maximal unramified 2-extension fixing the genus field of the quadratic field. Our method additionally can be used to determine the asymptotics of the unnormalized counting function, which we also do for unramified |$H_{8}$|-extensions. Furthermore, we propose that a similar normalization of the counting function is necessary to obtain finite moments when |$H_{8}$| is replaced by any 2-group |$G$|⁠.

中文翻译：

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二次场的H _8-扩展的分布

我们计算函数归一化的所有时间，这些归一化计数的是未加| $ H_ {8} $ | -二次字段的扩展，其中| $ H_ {8} $ | 是阶| $ 8 $ |⁠的四元数组，并表明此函数的值确定点的质量分布。作为对2组非阿贝尔Cohen-Lenstra启发式分析的结果，我们表明这暗示着，任何固定组作为最大无分支2扩展的Galois组的子组出现的概率为0，固定了二次场。我们的方法还可以用于确定非归一化计数函数的渐近性，我们也可以对|| $ H_ {8} $ | | | | | | | | | | | | | | | | | | | | | |-扩展名。此外，我们建议当| $ H_ {8} $ |时，为了获得有限矩，必须对计数函数进行类似的归一化。由任何2组| $ G $ |⁠代替。