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

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 $(A_h,T{r}_h)$ is locally $G$-isometric to $({\mathcal{O}}_{K}G,t_K)$ and hence defines a class in the unitary class group $\mbox{UCl}(\mathcal{O}_KG)$ of ${\mathcal{O}}_{K}G$. Here $T{r}_h$ denotes the trace of $K_h/K$ and $t_K$ the symmetric bilinear form on ${\mathcal{O}}_{K}G$ 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 $\mbox{UCl}(\mathcal{O}_KG)$.

Version 2; fixed a few errors and added more details in throughout the paper

References in corpus (1)

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

Cited by in corpus (1)

• On the relative Galois module structure of rings of integers in tame extensions