UC Berkeley

One goal of this thesis is to prove theorems that elucidate the Cohen-Lenstra-Martinet conjectures for the distributions of class groups of number fields, and further the understanding of their implications. We start by giving a simpler statement of the conjectures. We show that the probabilities that arise are inversely proportional to the number of automorphismsof structures slightly larger than the class groups. We find the moments of the Cohen-Lenstra-Martinet distributions and prove that the distributions are determined by their moments. In order to apply these conjectures to class groups of non-Galois fields, we prove a new theorem on the capitulation kernel (of ideal classes that become trivial in a larger field) to relate the class groups of non-Galois fields to the class groups of Galois fields. We then construct an integral model of the Hecke algebra of a finite group, show that it acts naturally on class groups of non-Galois fields, and prove that the Cohen-Lenstra-Martinet conjectures predict a distribution for class groups of non-Galois fields that involves the inverse of the number of automorphisms of the class group as a Hecke-module. The Cohen-Lenstra-Martinet Heuristics give a prediction for the distribution for the p-Sylow subgroups of the class groups of random Γ-number fields when p ∤ |Γ|. In this thesis, we prove several results on the distributions of the class groups for some p||Γ|, and show that the behaviour is qualitatively different than the predicted behaviour when p ∤ |Γ|. We do this by using genus theory and the invariant part of the class group to investigate the algebraic structure of the bad part of the class group. For general number fields, our result is conditional on a natural conjecture on field counting. For abelian or D4 fields, our result is unconditional.