On the distribution of class groups of number fields

We propose a modification of the predictions of the Cohen--Lenstra heuristic for class groups of number fields in the case where roots of unity are present in the base field. As evidence for this modified formula we provide a large set of computational data which show close agreement.Comment: 14 pages. To appear in Experimental Mat

Principalization algorithm via class group structure

For an algebraic number field K with 3-class group $Cl_3(K)$ of type (3,3), the structure of the 3-class groups $Cl_3(N_i)$ of the four unramified cyclic cubic extension fields $N_i$, $1\le i\le 4$, of K is calculated with the aid of presentations for the metabelian Galois group $G_3^2(K)=Gal(F_3^2(K) | K)$ of the second Hilbert 3-class field $F_3^2(K)$ of K. In the case of a quadratic base field $K=\mathbb{Q}(\sqrt{D})$ it is shown that the structure of the 3-class groups of the four $S_3$-fields $N_1,\ldots,N_4$ frequently determines the type of principalization of the 3-class group of K in $N_1,\ldots,N_4$. This provides an alternative to the classical principalization algorithm by Scholz and Taussky. The new algorithm, which is easily automatizable and executes very quickly, is implemented in PARI/GP and is applied to all 4596 quadratic fields K with 3-class group of type (3,3) and discriminant $-10^6<D<10^7$ to obtain extensive statistics of their principalization types and the distribution of their second 3-class groups $G_3^2(K)$ on various coclass trees of the coclass graphs G(3,r), $1\le r\le 6$, in the sense of Eick, Leedham-Green, and Newman.Comment: 33 pages, 2 figures, presented at the Joint CSASC Conference, Danube University, Krems, Austria, September 201

Distribution of the bad part of the class group

The Cohen-Lenstra-Martinet Heuristics gives a prediction of the distribution of $\operatorname{Cl}_K[p^\infty]$ whne $K$ runs over $\Gamma$-fields and $p\nmid|\Gamma|$. In this paper, we prove several results on the distribution of ideal class groups for some $p||\Gamma|$, and show that the behaviour is qualitatively different than what is predicted by the heuristics when $p\nmid|\Gamma|$.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 $D_4$-fields, our result is unconditional.Comment: All comments are welcome

Conjectures for distributions of class groups of extensions of number fields containing roots of unity

Cohen, Lenstra, and Martinet have given conjectures for the distribution of class groups of extensions of number fields, but Achter and Malle have given theoretical and numerical evidence that these conjectures are wrong regarding the Sylow $p$-subgroups of the class group when the base number field contains $p$th roots of unity. We give complete conjectures of the distribution of Sylow $p$-subgroups of class groups of extensions of a number field when $p$ does not divide the degree of the Galois closure of the extension. These conjectures are based on $q\rightarrow\infty$ theorems on these distributions in the function field analog and use recent work of the authors on explicitly giving a distribution of modules from its moments. Our conjecture matches many, but not all, of the previous conjectures that were made in special cases taking into account roots of unity.Comment: 35 page