Nonabelian Cohen–Lenstra moments
Nonabelian Cohen–Lenstra moments
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DOI：
10.1215/00127094-2018-0037
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发表时间：
2017-02
影响因子：
2.5
通讯作者：
M. Wood;Philip Matchett Wood
中科院分区：
文献类型：
--
作者：
M. Wood;Philip Matchett Wood
本文给出了有限群G的二次域的非分歧G-扩张的平均数的一个猜想。Cohen-Lenstra定理是我们的猜想在G是奇阶阿贝尔情形下的特殊化。我们证明了一个定理对功能领域模拟我们的猜想，并给出额外的动机，包括建设的提升不变量的unramified $G$-扩展，采取相同数量的值作为预测的平均值和参数使用马勒Bhargava原则。我们注意到，即使是$|G| $，需要对$\mathbb{Q}$中的单位根进行修正，当$G$是阿贝尔时，这是看不到的。
In this paper we give a conjecture for the average number of unramified $G$-extensions of a quadratic field for any finite group $G$. The Cohen-Lenstra heuristics are the specialization of our conjecture to the case that $G$ is abelian of odd order. We prove a theorem towards the function field analog of our conjecture, and give additional motivations for the conjecture including the construction of a lifting invariant for the unramified $G$-extensions that takes the same number of values as the predicted average and an argument using the Malle-Bhargava principle. We note that for even $|G|$, corrections for the roots of unity in $\mathbb{Q}$ are required, which can not be seen when $G$ is abelian.