scholarly journals Determining Fitting ideals of minus class groups via the equivariant Tamagawa number conjecture

2007 ◽  
Vol 143 (6) ◽  
pp. 1399-1426 ◽  
Author(s):  
Cornelius Greither
Keyword(s):  
Number Fields ◽  
Rational Numbers ◽  
Abelian Extension ◽  
Galois Module ◽  
Class Groups ◽  
Fitting Ideal ◽  
Tamagawa Number Conjecture

AbstractWe assume the validity of the equivariant Tamagawa number conjecture for a certain motive attached to an abelian extension K/k of number fields, and we calculate the Fitting ideal of the dual of clK− as a Galois module, under mild extra hypotheses on K/k. This builds on concepts and results of Tate, Burns, Ritter and Weiss. If k is the field of rational numbers, our results are unconditional.

scholarly journals THE FRACTIONAL GALOIS IDEAL FOR ARBITRARY ORDER OF VANISHING

2011 ◽  
Vol 07 (01) ◽  
pp. 87-99 ◽  
Author(s):  
PAUL BUCKINGHAM
Keyword(s):  
Class Group ◽  
Arbitrary Order ◽  
Number Fields ◽  
Simple Zero ◽  
Abelian Extension ◽  
Class Groups ◽  
Original Proof ◽  
Galois Modules ◽  
Fitting Ideal ◽  
Stickelberger Ideal

We propose a candidate, which we call the fractional Galois ideal after Snaith's fractional ideal, for replacing the classical Stickelberger ideal associated to an abelian extension of number fields. The Stickelberger ideal can be seen as gathering information about those L-functions of the extension which are non-zero at the special point s = 0, and was conjectured by Brumer to give annihilators of class-groups viewed as Galois modules. An earlier version of the fractional Galois ideal extended the Stickelberger ideal to include L-functions with a simple zero at s = 0, and was shown by the present author to provide class-group annihilators not existing in the Stickelberger ideal. The version presented in this paper deals with L-functions of arbitrary order of vanishing at s = 0, and we give evidence using results of Popescu and Rubin that it is closely related to the Fitting ideal of the class-group, a canonical ideal of annihilators. Finally, we prove an equality involving Stark elements and class-groups originally due to Büyükboduk, but under a slightly different assumption, the advantage being that we need none of the Kolyvagin system machinery used in the original proof.

scholarly journals THE EQUIVALENCE OF RUBIN'S CONJECTURE AND THE ETNC/LRNC FOR CERTAIN BIQUADRATIC EXTENSIONS

2013 ◽  
Vol 56 (2) ◽  
pp. 335-353 ◽  
Author(s):  
PAUL BUCKINGHAM
Keyword(s):  
Number Fields ◽  
Abelian Extension ◽  
Root Number ◽  
Tamagawa Number Conjecture

AbstractFor an abelian extension L/K of number fields, the Equivariant Tamagawa Number Conjecture (ETNC) at s = 0, which is equivalent to the Lifted Root Number Conjecture (LRNC), implies Rubin's Conjecture by work of Burns [3]. We show that, for relative biquadratic extensions L/K satisfying a certain condition on the splitting of places, Rubin's Conjecture in turn implies the ETNC/LRNC. We conclude with some examples.

scholarly journals Galois module structure of units in real biquadratic number fields

2004 ◽  
Vol 111 (2) ◽  
pp. 105-124 ◽  
Author(s):  
Marcin Mazur ◽  
Stephen V. Ullom
Keyword(s):  
Number Fields ◽  
Module Structure ◽  
Galois Module ◽  
Galois Module Structure

The p -Dimension of Class Groups of Number Fields

1970 ◽  
Vol 2 (Part_3) ◽  
pp. 525-529 ◽  
Author(s):  
I. Connell ◽  
D. Sussman
Keyword(s):  
Number Fields ◽  
Class Groups

scholarly journals A Note on Ideal Class Groups

1966 ◽  
Vol 27 (1) ◽  
pp. 239-247 ◽  
Author(s):  
Kenkichi Iwasawa
Keyword(s):  
Class Group ◽  
Cyclotomic Field ◽  
Theoretical Method ◽  
Number Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Algebraic Number Fields ◽  
Ideal Class Groups ◽  
The Ideal

In the first part of the present paper, we shall make some simple observations on the ideal class groups of algebraic number fields, following the group-theoretical method of Tschebotarew. The applications on cyclotomic fields (Theorems 5, 6) may be of some interest. In the last section, we shall give a proof to a theorem of Kummer on the ideal class group of a cyclotomic field.

scholarly journals Correction to: Stickelberger ideals and Fitting ideals of class groups for abelian number fields

2019 ◽  
Vol 374 (3-4) ◽  
pp. 2083-2088
Author(s):  
Masato Kurihara ◽  
Takashi Miura
Keyword(s):  
Number Fields ◽  
Class Groups

scholarly journals On class groups of random number fields

2020 ◽  
Vol 121 (4) ◽  
pp. 927-953
Author(s):  
Alex Bartel ◽  
Hendrik W. Lenstra
Keyword(s):  
Random Number ◽  
Number Fields ◽  
Class Groups
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