Iwasawa theory of Rubin-Stark units and class group

TL;DR

This paper extends Iwasawa theory results relating class groups and units, specifically Rubin-Stark units, in the setting of totally real fields and their cyclotomic extensions, including non semi-simple cases.

Contribution

It generalizes previous work by connecting characteristic ideals of class groups to exterior powers of units for certain characters in non semi-simple cases.

Findings

Established relations between class groups and Rubin-Stark units in new settings.

Extended Iwasawa theory results to non semi-simple cases.

Provided new insights into the structure of units and class groups in cyclotomic extensions.

Abstract

Let $K$ be a totally real number field of degree $r=[K:\mathbb{Q}]$ and let $p$ be an odd rational prime. Let $K_{\infty}$ denote the cyclotomic $\mathbb{Z}_{p}$-extension of $K$ and let $L_{\infty}$ be a finite extension of $K_{\infty}$, abelian over $K$. In this article, we extend results of \cite{Kazim109} relating characteristic ideal of the $\chi$-quotient of the projective limit of the ideal class groups to the $\chi$-quotient of the projective limit of the $r$-th exterior power of units modulo Rubin-Stark units, in the non semi-simple case, for some $\overline{\mathbb{Q}_{p}}$-irreductible characters $\chi$ of $\mathrm{Gal}(L_{\infty}/K_{\infty})$.