Seminar Notes on Open Questions in Iwasawa Theory - SNOQIT I: The $\Lambda[ G ]$-modules of Iwasawa theory II: Units and Kummer theory in Iwasawa extensions

TL;DR

This paper explores properties of $ ext{Lambda}$-modules in Iwasawa theory, introduces a new proof for Leopoldt and Gross conjectures in CM extensions, and advances Greenberg's conjecture by proving $ ext{lambda}^+ = 0$ in these fields.

Contribution

It provides a new, simplified proof of key conjectures in Iwasawa theory for CM extensions and establishes $ ext{lambda}^+ = 0$, advancing understanding of Iwasawa modules.

Findings

New proof of Leopoldt and Gross conjectures for CM extensions

Proved $ ext{lambda}^+ = 0$ in these fields

Partial elementary proofs of main facts using Iwasawa's linear space

Abstract

In Part I we review some specific properties of the $\Lambda$-modules in Iwasawa theory, which add structure to the general properties of Noetherian $\Lambda$-torsion modules. Part II deals with Kummer theory and gives a detailed construction of the Iwasawa linear space. This provides a new, simpler proof of the conjectures of Leopoldt and Gross for CM extensions. Using a construction of Thaine, we then prove that $\lambda^+ = 0$ in these fields, thus proving a part of Greenberg's conjecture - the fact $\mu^+ = 0$ still has to be shown. In the Appendices we give some elementary partial proofs of the main facts proved using Iwasawa's linear space. These two papers do not give proofs for non CM fields, and the reader interested in methods for dealing with this case is referred to the "$T and T^*$" paper on this arxive. This methods will be integrated in the Snoqit series.