On units generated by Euler systems

TL;DR

This paper investigates the relationship between Euler systems and cyclotomic units in cyclotomic fields, proving they generate the same module for any odd prime using Iwasawa theory techniques.

Contribution

It establishes that units from Euler systems and cyclotomic units generate the same z-module, extending understanding of their structure in cyclotomic fields.

Findings

Units from Euler systems and cyclotomic units generate the same z-module for any odd prime p.

Provides an Iwasawa theoretic proof based on Rubin's work on main conjectures.

Clarifies the generation of norm-compatible units by Euler systems in cyclotomic fields.

Abstract

In the context of cyclotomic fields, it is still unknown whether there exist Euler systems other than the ones derived from cyclotomic units. Nevertheless, we first give an exposition on how norm-compatible units are generated by any Euler system, following work of Coates. Then we prove that the units obtained from Euler systems and the cyclotomic units generate the same $\mathbb{Z}_{p}$-module for any odd prime $p$. The techniques adopted for the Iwasawa theoretic proof in latter part of this article originated in Rubin's work on main conjectures of Iwasawa theory.