On the Galois module structure of the square root of the inverse different in abelian extensions

TL;DR

This paper investigates the Galois module structure of the square root of the inverse different in weakly ramified abelian Galois extensions over number fields, showing that these classes form a subgroup under certain conditions.

Contribution

It proves that, for abelian Galois groups and suitable assumptions, the classes of the square root of the inverse different form a subgroup in the locally free class group.

Findings

Classes form a subgroup in the class group for abelian Galois groups.

The module is locally free over the group ring.

Results depend on weak ramification and specific assumptions.

Abstract

Let $K$ be a number field with ring of integers $\mathcal{O}_K$ and $G$ a finite group of odd order. If $K_h$ is a weakly ramified $G$-Galois $K$-algebra, then its square root $A_h$ of the inverse different is a locally free $\mathcal{O}_{K}G$-module and hence determines a class in the locally free class group $\mbox{Cl}(\mathcal{O}_KG)$ of $\mathcal{O}_KG$. We show that for $G$ abelian and under suitable assumptions, the set of all such classes is a subgroup of $\mbox{Cl}(\mathcal{O}_KG)$.