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REFLEXIVE IDEALS IN IWASAWA ALGEBRAS K. ARDAKOV, F. WEI AND J. J. ZHANG
 

Summary: REFLEXIVE IDEALS IN IWASAWA ALGEBRAS
K. ARDAKOV, F. WEI AND J. J. ZHANG
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 G of
G has no non-trivial two-sided reflexive ideals. Consequently, these conditions
imply that every nonzero normal element in G is a unit. We show that these
conditions hold in the case when G is an open subgroup of SL2(Zp) and p is
arbitrary. Using a previous result of the first author, we show that there are
only two prime ideals in G when G is a congruence subgroup of SL2(Zp): the
zero ideal and the unique maximal ideal. These statements partially answer
some questions asked by the first author and Brown.
0. Introduction
0.1. Motivation. The Iwasawa theory for elliptic curves in arithmetic geometry
provides the main motivation for the study of Iwasawa algebras G, for example
when G is a certain subgroup of the p-adic analytic group GL2(Zp) [CSS, Section 8].
Homological and ring-theoretic properties of these Iwasawa algebras are useful for
understanding the structure of the Pontryagin dual of Selmer groups [OV, V3] and
other modules over the Iwasawa algebras. Several recent papers [A, AB1, AB2, V1,
V2] are devoted to ring-theoretic properties of the Iwasawa algebras. One central
question in this research direction is whether there are any non-trivial prime ideals

  

Source: Ardakov, Konstantin - School of Mathematical Sciences, University of Nottingham

 

Collections: Mathematics