Iwasawa theory of Rubin-Stark units and class groups

Abstract

Let K be a totally real number field of degree r = [K : Q] and let p be an odd rational prime. Let K∞ denote the cyclotomic Zp-extension of K and let L∞ be a finite extension of K∞, abelian over K. In this article, we extend results of [Bü 09] relating characteristic ideal of the χ-quotient of the projective limit of the ideal class groups to the χ-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 Qp-irreductible characters χ of Gal(L∞/K∞). × p be a non-trivial Q p -irreducible character of ∆. Let O := Z p [χ] be the ring generated by the values of χ over Z p and let O(χ) denote the ring O on which ∆ acts via χ. For any Z p [∆]-module M , we define the χ-quotient M χ by M χ := M ⊗ Zp[∆] O(χ).

On Iwasawa Theory of Rubin–stark Units and Narrow Class Groups

“…Since L n is totally real, by isomorphism (10), we get Tor O (H 1 (G K n , , T)) = 0. Therefore, the exact sequence (9) shows that Tor O (H 1 + (G K n , , T)) = 0; hence,…”

“…Proof of Theorem 1.1. We will take the notations and the conventions of [9]. In particular, the construction of the group of Rubin-Stark units [9, Definition 4.5] goes on the same lines.…”

“…For simplicity of notation, we let ε n stand for the Rubin-Stark element ε n, h,T for RS(L n /K, S L n , T , r). Remark 4.2 in [9] shows that ε n can be written as…”

“…In [9], for a fixed odd rational prime p, we used the theory of Euler systems to bound the size of the χ -quotient of the p-class groups by the characteristic ideal of the χ -quotient of the rth exterior power of units modulo Rubin-Stark units, in the non-semi-simple case, thus extending the results of [4].…”