Ideal class groups of CM-fields with non-cyclic galois action

Masato Kurihara, Takashi Miura

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

Suppose that L/k is a finite and abelian extension such that k is a totally real base field and L is a CM-field. We regard the ideal class group Cl L, of L as a Gal(L/k)-module. As a sequel of the paper [8] by the first author, we study a problem whether the Stickelberger element for L/k times the annihilator ideal of the roots of unity in L is in the Fitting ideal of ClL, and also a problem whether it is in the Fitting ideal of the Pontrjagin dual (ClL)v. We systematically construct extensions L/k for which these properties do not hold, and also give numerical examples.

Original languageEnglish
Pages (from-to)411-439
Number of pages29
JournalTokyo Journal of Mathematics
Volume35
Issue number2
DOIs
Publication statusPublished - 2012 Dec

Fingerprint

CM-field
Ideal Class Group
Galois
Annihilator
Roots of Unity
Numerical Examples
Module

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Ideal class groups of CM-fields with non-cyclic galois action. / Kurihara, Masato; Miura, Takashi.

In: Tokyo Journal of Mathematics, Vol. 35, No. 2, 12.2012, p. 411-439.

Research output: Contribution to journalArticle

@article{99b4c2defc93497584dff7a54e854928,
title = "Ideal class groups of CM-fields with non-cyclic galois action",
abstract = "Suppose that L/k is a finite and abelian extension such that k is a totally real base field and L is a CM-field. We regard the ideal class group Cl L, of L as a Gal(L/k)-module. As a sequel of the paper [8] by the first author, we study a problem whether the Stickelberger element for L/k times the annihilator ideal of the roots of unity in L is in the Fitting ideal of ClL, and also a problem whether it is in the Fitting ideal of the Pontrjagin dual (ClL)v. We systematically construct extensions L/k for which these properties do not hold, and also give numerical examples.",
author = "Masato Kurihara and Takashi Miura",
year = "2012",
month = "12",
doi = "10.3836/tjm/1358951328",
language = "English",
volume = "35",
pages = "411--439",
journal = "Tokyo Journal of Mathematics",
issn = "0387-3870",
publisher = "Publication Committee for the Tokyo Journal of Mathematics",
number = "2",

}

TY - JOUR

T1 - Ideal class groups of CM-fields with non-cyclic galois action

AU - Kurihara, Masato

AU - Miura, Takashi

PY - 2012/12

Y1 - 2012/12

N2 - Suppose that L/k is a finite and abelian extension such that k is a totally real base field and L is a CM-field. We regard the ideal class group Cl L, of L as a Gal(L/k)-module. As a sequel of the paper [8] by the first author, we study a problem whether the Stickelberger element for L/k times the annihilator ideal of the roots of unity in L is in the Fitting ideal of ClL, and also a problem whether it is in the Fitting ideal of the Pontrjagin dual (ClL)v. We systematically construct extensions L/k for which these properties do not hold, and also give numerical examples.

AB - Suppose that L/k is a finite and abelian extension such that k is a totally real base field and L is a CM-field. We regard the ideal class group Cl L, of L as a Gal(L/k)-module. As a sequel of the paper [8] by the first author, we study a problem whether the Stickelberger element for L/k times the annihilator ideal of the roots of unity in L is in the Fitting ideal of ClL, and also a problem whether it is in the Fitting ideal of the Pontrjagin dual (ClL)v. We systematically construct extensions L/k for which these properties do not hold, and also give numerical examples.

UR - http://www.scopus.com/inward/record.url?scp=84890189154&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84890189154&partnerID=8YFLogxK

U2 - 10.3836/tjm/1358951328

DO - 10.3836/tjm/1358951328

M3 - Article

AN - SCOPUS:84890189154

VL - 35

SP - 411

EP - 439

JO - Tokyo Journal of Mathematics

JF - Tokyo Journal of Mathematics

SN - 0387-3870

IS - 2

ER -