The p-adic group ring of SL_2(p^f)

TL;DR

This paper proves that the basic order of the p-adic group ring of SL_2 over a finite field can be uniquely identified among similar orders using ring-theoretic properties, confirming a conjecture by Nebe.

Contribution

It demonstrates the unique recognition and lifting of the basic order of the p-adic group ring of SL_2, confirming Nebe's conjecture.

Findings

The order $ ext{Z}_p[ ext{zeta}_{p^f-1}] ext{SL}_2(p^f)$ can be uniquely characterized by ring-theoretic properties.

The reduction modulo p of the order is isomorphic to $ ext{F}_{p^f} ext{SL}_2(p^f)$ and lifts uniquely under certain conditions.

The result confirms Nebe's conjecture regarding the basic order of the p-adic group ring.

Abstract

In this article we show that the $\Z_p[\zeta_{p^f-1}]$-order $\Z_p[\zeta_{p^f-1}]\SL_2(p^f)$ can be recognized among those orders whose reduction modulo $p$ is isomorphic to $\F_{p^f}\SL_2(p^f)$ using only ring-theoretic properties (in other words we show that $\F_{p^f}\SL_2(p^f)$ lifts uniquely to a $\Z_p[\zeta_{p^f-1}]$-order, provided certain reasonable conditions are imposed on the lift). This proves a conjecture made by Nebe concerning the basic order of $\Z_p[\zeta_{p^f-1}]\SL_2(p^f)$.