Completed group algebras of abelian-by-procyclic groups

Abstract

Let p be prime, let G be a p-valuable group, isomorphic to Zpd⋊Zp, and let k be a finite field of characteristic p. We will prove that all faithful prime ideals of the completed group algebra kG are controlled by Z(G), and a complete decomposition for Spec(kG) will follow. The principal technique we employ will be to study the convergence of Mahler expansions for inner automorphisms.