Construction of a cyclic $p$-extension of number fields whose unit group has prescribed Galois module structure

Abstract

For any finite cyclic $p$-group $G$, we will show that every $\mathbb{Z}_p$-torsion free finitely generated $\mathbb{Z}_p[G]$-module appears as $\mathcal{O}_K^\times\otimes_{\mathbb{Z}}\mathbb{Z}_p$ up to $\mathbb{Z}_p[G]$-free direct summands for a certain $G$-extension $K/k$ of number fields.