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

TL;DR

This paper proposes new conjectures for the distribution of Sylow p-subgroups of class groups in number field extensions containing roots of unity, supported by analogies with function field results and recent module distribution work.

Contribution

It introduces complete conjectures for class group distributions when p does not divide the Galois closure degree, extending previous conjectures to cases with roots of unity.

Findings

Conjectures align with many previous special case conjectures.

Supports conjectures with analogies from function field theory.

Uses recent work on module distributions from moments.

Abstract

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.