Rarity of pseudo-null Iwasawa modules for $p$-adic Lie extensions

TL;DR

This paper establishes a precise criterion for when the $p$-ramified Iwasawa module is pseudo-null in $p$-adic Lie extensions of totally real fields, revealing that such cases are extremely uncommon.

Contribution

It provides a necessary and sufficient condition for pseudo-nullity of the $p$-ramified Iwasawa module, advancing understanding of its rarity in specific number field extensions.

Findings

Pseudo-nullity is very rare in the studied extensions.

The paper offers a complete criterion for pseudo-nullity.

Applications include the minus component of unramified Iwasawa modules for CM-fields.

Abstract

In this paper, we obtain a necessary and sufficient condition for the pseudo-nullity of the $p$-ramified Iwasawa module for $p$-adic Lie extension of totally real fields. It is applied to answer the corresponding question for the minus component of the unramified Iwasawa module for CM-fields. The results show that the pseudo-nullity is very rare.