On realizable Galois module classes by the inverse different

Abstract

Let k be a number field and O_k its ring of integers. Let Γ be a finite group. Let $\mathcal{M}$ be a maximal O_k-order in the semi-simple algebra $k[\Gamma ]$ containing $O_k[\Gamma ].$ Let $Cl(O_k[\Gamma ])$ (resp. $Cl(\mathcal{M})$) be the locally free classgroup of $O_k[\Gamma ]$ (resp. $\mathcal{M}$). We denote by $\mathcal{R}({\mathcal{D}}^{-1},O_k[\Gamma ])$ (resp. $\mathcal{R}({\mathcal{D}}^{-1},\mathcal{M})$) the set of classes c in $Cl(O_k[\Gamma ])$ (resp. $Cl(\mathcal{M})$) such that there exists a tamely ramified extension N/k, with Galois group isomorphic to Γ (Γ-extension) and the class of ${\mathcal{D}}_{N/k}^{-1}$ (resp. $\mathcal{M}{\otimes }_{O_k[\Gamma ]}{\mathcal{D}}_{N/k}^{-1}$) is equal to c, where ${\mathcal{D}}_{N/k}$ is the different of N/k and ${\mathcal{D}}_{N/k}^{-1}$ its inverse different. We say that $\mathcal{R}({\mathcal{D}}^{-1},O_k[\Gamma ])$ (resp. $\mathcal{R}({\mathcal{D}}^{-1},\mathcal{M})$) is the set of realizable Galois module classes by the inverse different. In the present article, combining some of our published results, and a result due to A. Fröhlich giving a link between the Galois module class of the ring of integers of a tamely ramified $\Gamma$-extension and that of its inverse different, we explicitly determine $\mathcal{R}({\mathcal{D}}^{-1},O_k[\Gamma ])$ (resp. $\mathcal{R}({\mathcal{D}}^{-1},\mathcal{M})$) for several groups $\Gamma$ and show that it is a subgroup of $Cl(O_k[\Gamma ])$ (resp. $Cl(\mathcal{M})$). In addition, we determine the set of the Steinitz classes of ${\mathcal{D}}_{N/k}^{-1},$ N/k runs through the set of tamely ramified Γ-extension of k, and prove that is a subgroup of Cl(k), also for several groups $\Gamma .$

Keywords:

• Different
• Galois module structure
• inverse different
• Lagrange resolvent
• locally free classgroups
• maximal order
• realizable classes
• rings of integers
• Steinitz classes
• Stickelberger ideal

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 11R33

Acknowledgments

The authors thank the referee for his very quick and efficient work despite the various difficulties and limitations due to Coronavirus. B. Sodaïgui is very grateful to the CNRS which granted him a delegation during the academic year 2019–2020.