Reflexive Ideals in Iwasawa Algebras

TL;DR

This paper investigates the structure of Iwasawa algebras for certain p-adic groups, establishing conditions under which these algebras lack non-trivial reflexive ideals and have a simple prime ideal structure.

Contribution

It provides new sufficient conditions on groups and primes ensuring the absence of non-trivial reflexive ideals in Iwasawa algebras, and characterizes their prime ideal structure in specific cases.

Findings

No non-trivial two-sided reflexive ideals under certain conditions.

Every nonzero normal element in the algebra is a unit.

Only two prime ideals in the algebra for certain subgroups.

Abstract

Let $G$ be a torsionfree compact $p$-adic analytic group. We give sufficient conditions on $p$ and $G$ which ensure that the Iwasawa algebra $\Omega_G$ of $G$ has no non-trivial two-sided reflexive ideals. Consequently, these conditions imply that every nonzero normal element in $\Omega_G$ is a unit. We show that these conditions hold in the case when $G$ is an open subgroup of $\SL_2(\Zp)$ and $p$ is arbitrary. Using a previous result of the first author, we show that there are only two prime ideals in $\Omega_G$ when $G$ is a congruence subgroup of $\SL_2(\Zp)$: the zero ideal and the unique maximal ideal. These statements partially answer some questions asked by the first author and Brown.