Class groups and unramifed extensions of number fields

研究数域非分歧扩张的分布。对于阿贝尔扩张，这只是类群的分布。申请人最近发现了证据，证明Cohen-Lenstra启发法对于阶数不与基域中的单位根的阶数互素的类群的部分失效。这个项目的一个主要目的是从全局函数场的情况下获得具体的数值预测。这涉及到推导出明确的公式，在有限辛群的剩余环与给定的本征空间的元素的数量。在非交换的情况下，我们建议从群论的角度通过相应的扩张问题来研究小导出长度的高阶群，并通过实验的方法获得关于非交换简单伽罗瓦群的非分歧扩张的第一个见解。在历史上，类域理论、Cohen-Lenstra启发式和申请人关于伽罗瓦群和类群的分布的说明是基于大域集合的统计的例子和表格而发展的。相应地，除了理论部分，我们的项目有大量的计算和实验方面。

The proposed research concerns the distribution of unramified extensions of number fields. For abelian extensions this is just the distribution of class groups. The applicant has recently found evidence that the Cohen–Lenstra heuristic breaks down for the part of the class group of order not prime to the order of the roots of unity in the base field. One major aim of this project is to obtain concrete numerical predictions from the global function field case. This involves deriving explicit formulas for the number of elements in finite symplectic groups over residue rings with given eigenspaces. In the non-abelian case we propose to study higher class groups of small derived length from a group theoretical point of view via the corresponding extension problems, and also to obtain first insights on unramified extensions with non-abelian simple Galois group by experimental methods. Historically, class field theory, the Cohen–Lenstra heuristic and the conjectures by the applicant on distributions of Galois groups and class groups were developed on the basis of examples and tables of statistics for large sets of fields. Correspondingly, besides theoretical parts our project has substantial computational and experimental aspects.