Euler systems and special units

✍ Scribed by Soogil Seo

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 199 KB
• Volume: 109
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2004.06.010

✦ Synopsis

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 n K ) be a sequence of decreasing subgroups S n K (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 n F /S n+1 F isomorphic as Z[Gal(F /Q)] modules? Let be the set of p-adic valued Dirichlet characters of Gal(F /Q). Let S F , C F and Cl F be the -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 n F ) /(S n+1 F ) is less than or equal to the Z/pZ-rank of Cl F 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 ∈ , we have S F = C F ⇐⇒ Cl F is a cyclic group.