Euler systems and special units

Let F be a number field. We investigate the group of Rubinʹs special units, SF defined over F. The group of special units is a subgroup of the group of global units containing the group of Sinnottʹs cyclotomic units, CF of F. It plays an important role in studying the ideal class group of F. Let be a sequence of decreasing subgroups (defined in Section 2) of the group of global units of any real abelian field K which lie between Rubinʹs special units and the circular units of K. Motivated by a question of whether the group of special units equals the group of cyclotomic units, which is stated by Rubin (Invent. Math. 89 (1987) 511), we propose the following question which relates the group structure of the ideal class group with the group structure of units modulo special units. Are and isomorphic as Z[Gal(F/Q)] modules? Let Ξ be the set of p-adic valued Dirichlet characters of Gal(F/Q). Let and be the χ-eigenspaces of and ClF Zp respectively. Using Euler system methods and Thaineʹs results we obtain that the Z/pZ-rank of is less than or equal to the Z/pZ-rank of with some inequalities on the cardinalities of both sides. This gives us the following corollary. If , then for all χ Ξ, we have is a cyclic group.