Iwasawa theory of Rubin-Stark units and class groups

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 Büyükboduk (Compos Math 145(5):1163–1195, 2009) 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 $\overline{\mathbb {Q}_{p}}$-irreducible characters $\chi$ of $\mathrm {Gal}(L_{\infty }/K_{\infty })$.