On Iwasawa theory of Rubin-Stark units and narrow class groups

Abstract

Let $K$ be a totally real number field of degree $r$. Let $K_{\infty }$ denote the cyclotomic ${\mathbb{Z}}_2$-extension of $K$ and let $L_{\infty }$ be a finite extension of $K_{\infty }$, abelian over $K$. The goal of this paper is to compare the characteristic ideal of the $\chi$-quotient of the projective limit of the narrow class groups to the $\chi$-quotient of the projective limit of the $r$-th exterior power of totally positive units modulo a subgroup of Rubin-Stark units, for some $\overline{{\mathbb{Q}}_2}$-irreducible characters $\chi$ of $\mathrm{Gal}(L_{\infty }/K_{\infty })$.