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

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 $\Z_p[\zeta_{p^f-1}]\SL_2(p^f)$.