A conductor formula for completed group algebras

TL;DR

This paper extends Jacobinski's conductor formula to completed group algebras of 1-dimensional p-adic Lie groups, providing a comprehensive description of the central conductor in this setting.

Contribution

It introduces a new conductor formula for completed group algebras of p-adic Lie groups, generalizing classical results for finite groups.

Findings

Derived a conductor formula for $ ext{O}[[G]]$ where $G$ is a 1-dimensional p-adic Lie group.

Connected the formula to character theory of $G$ and classical conductor concepts.

Discussed implications for algebraic and number-theoretic structures.

Abstract

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