Class groups and unramifed extensions of number fields

本文主要研究数域的非分枝扩展的分布。对于阿贝尔扩展，这就是类群的分布。申请人最近发现的证据表明，科恩-伦斯特拉启发式分解为非素数阶的类群在基域的统一根的阶。该项目的一个主要目的是从全局函数场情况中获得具体的数值预测。这涉及到在给定特征空间的残环上导出有限辛群中元素个数的显式公式。在非阿贝尔情况下，通过相应的可拓问题，从群理论的角度研究了派生长度较小的高类群，并通过实验方法首次获得了非阿贝尔简单伽罗瓦群的非分支可拓。历史上，类场论、Cohen-Lenstra启发式以及申请人对伽罗瓦群和类群分布的猜想都是在大场集的实例和统计表的基础上发展起来的。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.