Prime ideals of Iwasawa algebras over solvable groups

Thesis

In this thesis, we will explore the non-commutative Iwasawa algebra OG of a uniform pro-p group G over the ring of integers O of a finite extension K of Q_p. Our aim is to explore the prime ideal structure of OG, and our ultimate goal is to prove that all prime ideals have a standard form for which we have a complete classification. We will focus on the case where G is solvable, and we will divide the problem into two cases, prime ideals containing p and prime ideals not containing p.

In the former case, we may quotient out by p and study the mod-p Iwasawa algebra kG, where k is the residue field of K. For G nilpotent, the classification in this case was completed by Ardakov, using the theory of Mahler expansions of continuous automorphisms of G. In the first half of this thesis, we will recap this theory, and show how it can be generalised to complete the classification for prime ideals in kG for G an abelian-by-procyclic group.

In the latter case, we may tensor with K and study the rational Iwasawa algebra KG. The theory of Mahler expansions ultimately fails in this case, so we will instead explore methods involving p-adic representation theory. We focus on the case where G is nilpotent abelian-by-procyclic, and we will complete the classification for faithful primitive ideals of KG in this case for p > 2. Our main technique will be to study the representation theory of the completed enveloping algebra of the associated Lie algebra L of G, and to describe all primitive ideals of this algebra using a class of canonical representations.

Authors: Jones, A

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Rigid geometry; Valuation theory; Two-sided ideals; Iwasawa algebras; Compact p-adic Lie groups
• Subjects: Lie theory; Non-commutative algebra; Group theory; Representation theory