Let F be a number field. We investigate the group of Rubin's special units, SF 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, CF of F. It plays an important role in studying the ideal class group of F. Let (SKn) be a sequence of decreasing subgroups SKn (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 ClF and n≥0 SFn/SFn+1 isomorphic as ℤ[Gal(F/ℚ)] modules? Let Ξ be the set of p-adic valued Dirichlet characters of Gal (F/ℚ). Let SFχ CF and ClFχ be the χ-eigenspaces of SF ⊗ ℤp, CF ⊗ ℤp and ClF ⊗ ℤp respectively. Using Euler system methods and Thaine's results we obtain that the ℤ/pℤ-rank of n≥0 (SFn) χ/(SFn+1)χ is less than or equal to the ℤ/pℤ-rank of ClFχ with some inequalities on the cardinalities of both sides. This gives us the following corollary. If p (2[F: ℚ], hF), then for all χ ∈ Ξ, we have SFχ=CFχ ⇔ ClFχ is a cyclic group.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory