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 Buyukboduk (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 \(\mathrm {Gal}(L_{\infty }/K_{\infty })\).