On the realizable classes of the square root of the inverse different in the unitary class group

TL;DR

This paper investigates the classes in the unitary class group arising from the square root of the inverse different in weakly ramified Galois extensions, showing that a certain subset forms a subgroup.

Contribution

It characterizes the realizable classes of the square root of the inverse different in the unitary class group and proves that a subset of these classes forms a subgroup.

Findings

The classes are locally G-isometric to the standard form under weak ramification.

A subset of these classes constitutes a subgroup of the unitary class group.

The study enhances understanding of the structure of classes associated with Galois algebras.

Abstract

Let $K$ be a number field with ring of integers $\mathcal{O}_K$ and let $G$ be a finite abelian group of odd order. Given a $G$-Galois $K$-algebra $K_h$, let $A_h$ denote its square root of the inverse different, which exists by Hilbert's formula. If $K_h/K$ is weakly ramified, then the pair $(A_h,Tr_h)$ is locally $G$-isometric to $(\mathcal{O}_KG,t_K)$ and hence defines a class in the unitary class group $\mbox{UCl}(\mathcal{O}_KG)$ of $\mathcal{O}_KG$. Here $Tr_h$ denotes the trace of $K_h/K$ and $t_K$ the symmetric bilinear form on $\mathcal{O}_KG$ for which $t_K(s,t)=\delta_{st}$ for all $s,t\in G$. We study the collection of all such classes and show that a subset of them is in fact a subgroup of $\mbox{UCl}(\mathcal{O}_KG)$.