### Abstract

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_{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 _{n≥0} S_{F}^{n}/S_{F}^{n+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_{F}^{n}) ^{χ}/(S_{F}^{n+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.

Original language | English |
---|---|

Pages (from-to) | 59-68 |

Number of pages | 10 |

Journal | Journal of Number Theory |

Volume | 109 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2004 Nov 1 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Journal of Number Theory*,

*109*(1), 59-68. https://doi.org/10.1016/j.jnt.2004.06.010

}

*Journal of Number Theory*, vol. 109, no. 1, pp. 59-68. https://doi.org/10.1016/j.jnt.2004.06.010

**Euler systems and special units.** / Seo, Soogil.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Euler systems and special units

AU - Seo, Soogil

PY - 2004/11/1

Y1 - 2004/11/1

N2 - 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.

AB - 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.

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U2 - 10.1016/j.jnt.2004.06.010

DO - 10.1016/j.jnt.2004.06.010

M3 - Article

VL - 109

SP - 59

EP - 68

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 1

ER -