Class groups and unramifed extensions of number fields

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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.