Centres of skewfields and completely faithful Iwasawa modules

TL;DR

This paper investigates the structure of Iwasawa algebras associated with certain p-adic groups, showing their skewfields have trivial centers and classifying prime ideals, with applications to elliptic curve Selmer groups.

Contribution

It proves the triviality of the center of the skewfield of fractions of Iwasawa algebras for split semisimple Lie algebra groups and classifies prime c-ideals, linking algebraic properties to Selmer groups.

Findings

The skewfield of fractions of mbda_H has trivial center.

Prime c-ideals in mbda_G are classified.

Finitely generated torsion modules are completely faithful iff they lack central torsion.

Abstract

Let H be a torsionfree compact p-adic analytic group whose Lie algebra is split semisimple. We show that the quotient skewfield of fractions of the Iwasawa algebra \Lambda_H of H has trivial centre and use this result to classify the prime c-ideals in the Iwasawa algebra \Lambda_G of G := H \times \Zp. We also show that a finitely generated torsion \Lambda_G-module having no non-zero pseudo-null submodule is completely faithful if and only if it is has no central torsion. This has an application to the study of Selmer groups of elliptic curves.