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.