Distribution of the bad part of class groups

TL;DR

This paper investigates the distribution of ideal class groups in number fields, revealing deviations from Cohen-Lenstra-Martinet heuristics when the prime divides the group order, using genus theory and invariant parts.

Contribution

It provides new results on class group distributions for primes dividing the group order, contrasting with existing heuristics, with some results unconditional for specific field types.

Findings

Distribution differs from heuristics when p divides |Γ|

Uses genus theory and invariant class group parts

Results are unconditional for abelian and D4-fields

Abstract

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.