Invariant part of class group and distribution of relative class group

TL;DR

This paper extends classical results on the invariant part of class groups to relative class groups, providing new statistical insights into their distribution in abelian and sextic number field extensions.

Contribution

It generalizes the invariant class group analysis to relative class groups and establishes infinite moments for Sylow p-subgroups in specific number field extensions.

Findings

Infinite Cp-moments for Sylow p-subgroups in abelian extensions.

Infinite C2-moments for Sylow 2-subgroups in sextic fields with A4-closure.

Extension of classical class group results to relative class groups.

Abstract

We generalize the work of Roquette and Zassenhaus on the invariant part of the class groups to the relative class groups. Thereby, we can show some statistical results as follows. For abelian extensions over a fixed number field K, we show infinite Cp-moments for the Sylow p-subgroup of the relative class group when p divides the degree of the extension. For sextic number fields with A4-closure, we can show infinite C2-moments for the Sylow 2-subgroup of the relative class group when the extensions run over a fixed Galois cubic field.