Euler systems and special units

[摘要] 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, C-F of F. It plays an important role in studying the ideal class group of F. Let (S-K(n)) be a sequence of decreasing subgroups S-K(n) (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 circle plus(ngreater than or equal to0) S-F(n)/S-F(n+1) isomorphic as Z[Gal(F/Q)] modules? Let Xi be the set of p-adic valued Dirichlet characters of Gal(F/Q). Let S-F(chi), C-F(chi) and Cl-F(chi) be the chi-eigenspaces Of S-F circle times Z(P); C-F circle times Z(P) and Cl-F circle times Z(P) respectively. Using Euler system methods and Thaine's results we obtain that the Z/pZ-rank of circle plus(ngreater than or equal to0)(S-F(n))(chi)/(S-F(n+1))(chi), is less than or equal to the Z/pZ-rank of Cl-F(chi) with some inequalities on the cardinalities of both sides. This gives us the following corollary. If p inverted iota (2[F : Q], h(F)), then for all chi is an element of Xi, we have S-F(chi) = C-F(chi) double left right arrow Cl-F(chi) is a cyclic group. (C) 2004 Elsevier Inc. All rights reserved.

[发布日期] 2004-11-01