A conductor formula for completed group algebras

Abstract

Let $\mathfrak{o}$ be the ring of integers in a finite extension of ${\mathbb{Q}}_p$. If $G$ is a finite group and $\Gamma$ is a maximal $\mathfrak{o}$-order containing the group ring $\mathfrak{o}[G]$, Jacobinski's conductor formula gives a complete description of the central conductor of $\Gamma$ into $\mathfrak{o}[G]$ in terms of characters of $G$. We prove a similar result for completed group algebras $\mathfrak{o}[[G]]$, where $G$ is a $p$-adic Lie group of dimension 1. We will also discuss several consequences of this result.