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Here is the list of publications and preprints:
Abstract: The Cohen-Lenstra-Martinet Heuristics gives a prediction of the distribution of ClK[p∞] whne K runs over Γ-fields and p∤|Γ|. In this paper, we prove several results on the distribution of ideal class groups for some p||Γ|, and show that the behaviour is qualitatively different than what is predicted by the heuristics 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 class group. For general number fields, our result is conditional on a natural conjecture on counting fields. For abelian or D4-fields, our result is unconditional.
Abstract: The goal of this paper is to further the understanding of the Cohen-Lenstra-Martinet conjectures for the distributions of class groups of number fields. We start by giving a simpler statement of the conjectures. We show that the probabilities that arise are inversely proportional the to number of automorphisms of 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.