Completed Group Algebras of Abelian-by-procyclic Groups

Let p be a prime, let G be a p-valuable, abelian-by-procyclic group, and let k be a field of characteristic p. We will prove that all faithful prime ideals of the completed group algebra kG are controlled by the centre of 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.