Galois module structure of the square root of the inverse different over maximal orders

TL;DR

This paper investigates the Galois module structure of the square root of the inverse different in number field extensions with odd order Galois groups, establishing conditions under which certain classes are equal after scalar extension.

Contribution

It proves that for abelian Galois groups, the set of all $A$-realizable classes coincides with the tame $A$-realizable classes after scalar extension to the maximal order.

Findings

Equality of $A$-realizable and tame $A$-realizable classes for abelian groups after scalar extension.

Conditions under which the class of the square root of the inverse different is locally free.

Extension of known results to broader classes of Galois modules.

Abstract

Let $K$ be a number field with ring of integers $\mathcal{O}_K$ and let $G$ be a \mbox{finite group of} odd order. Given a $G$-Galois $K$-algebra $K_h$, let $A_h$ be the square root of the inverse different of $K_h/K$, which exists by Hilbert's formula. If $K_h/K$ is weakly ramified, then $A_h$ is locally free over $\mathcal{O}_{K}G$ by a result of B. Erez, in which case it determines a class in the locally free class group $\mbox{Cl}(\mathcal{O}_KG)$ of $\mathcal{O}_KG$. Such a class in $\mbox{Cl}(\mathcal{O}_KG)$ is said to be $A$-realizable, and tame $A$-realizable if $K_h/K$ is tame. Let $\mathcal{A}(\mathcal{O}_KG)$ and $\mathcal{A}^t(\mathcal{O}_KG)$ denote the sets of all $A$-realizable classes and tame $A$-realizable classes, respectively. For $G$ abelian, we will show that the two sets $\mathcal{A}(\mathcal{O}_KG)$ and $\mathcal{A}^t(\mathcal{O}_KG)$ are equal when extended scalars to the maximal order in $KG$.