Euler systems and special units

Let F be a number field. We investigate the group of Rubin's special units, S_F defined over F. The group of special units is a subgroup of the group of global units containing the group of Sinnott's cyclotomic units, C_F of F. It plays an important role in studying the ideal class group of F. Let (S_Kⁿ) be a sequence of decreasing subgroups S_Kⁿ (defined in Section 2) of the group of global units of any real abelian field K which lie between Rubin's 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 Cl_F and _n≥0 S_Fⁿ/S_Fⁿ⁺¹ isomorphic as ℤ[Gal(F/ℚ)] modules? Let Ξ be the set of p-adic valued Dirichlet characters of Gal (F/ℚ). Let S_F^χ C_F and Cl_F^χ be the χ-eigenspaces of S_F ⊗ ℤ_p, C_F ⊗ ℤ_p and Cl_F ⊗ ℤ_p respectively. Using Euler system methods and Thaine's results we obtain that the ℤ/pℤ-rank of _n≥0 (S_Fⁿ) ^χ/(S_Fⁿ⁺¹)^χ is less than or equal to the ℤ/pℤ-rank of Cl_F^χ with some inequalities on the cardinalities of both sides. This gives us the following corollary. If p (2[F: ℚ], h_F), then for all χ ∈ Ξ, we have S_F^χ=C_F^χ ⇔ Cl_F^χ is a cyclic group.