ℓ-torsion bounds for the class group of number fields with an ℓ-group as Galois group

2021 ◽  
Author(s):  
Jürgen Klüners ◽  
Jiuya Wang
Keyword(s):  
Galois Group ◽  
Class Group ◽  
Number Fields

scholarly journals On -invariants attached to cyclic cubic number fields

2015 ◽  
Vol 18 (1) ◽  
pp. 684-698
Author(s):  
Daniel Delbourgo ◽  
Qin Chao
Keyword(s):  
Power Series ◽  
Zeta Function ◽  
Galois Group ◽  
Prime Number ◽  
Class Number ◽  
Class Group ◽  
Number Fields ◽  
Cubic Number Fields

We describe an algorithm for finding the coefficients of $F(X)$ modulo powers of $p$, where $p\neq 2$ is a prime number and $F(X)$ is the power series associated to the zeta function of Kubota and Leopoldt. We next calculate the 5-adic and 7-adic ${\it\lambda}$-invariants attached to those cubic extensions $K/\mathbb{Q}$ with cyclic Galois group ${\mathcal{A}}_{3}$ (up to field discriminant ${<}10^{7}$), and also tabulate the class number of $K(e^{2{\it\pi}i/p})$ for $p=5$ and $p=7$. If the ${\it\lambda}$-invariant is greater than zero, we then determine all the zeros for the corresponding branches of the $p$-adic $L$-function and deduce ${\rm\Lambda}$-monogeneity for the class group tower over the cyclotomic $\mathbb{Z}_{p}$-extension of $K$.Supplementary materials are available with this article.

Cyclotomic Galois module structure and the second Chinburg invariant

1995 ◽  
Vol 117 (1) ◽  
pp. 57-82
Author(s):  
Victor P. Snaith
Keyword(s):  
Galois Group ◽  
Group Ring ◽  
Galois Extension ◽  
Prime Power ◽  
Class Group ◽  
Number Fields ◽  
Affirmative Answer ◽  
Module Structure ◽  
Integral Group ◽  
Galois Module

AbstractWe study the second Chinburg invariant of a Galois extension of number fields. The Chinburg invariant lies in the class-group of the integral group-ring of the Galois group of the extension. A procedure is given whereby to evaluate the invariant in the case of the real cyclotomic case of regular prime power conductor and their subextensions of p-power degree. The invariant is shown to be zero in the latter cases, which yields new examples giving an affirmative answer to a question of Chinburg ([1], p. 358) which has come to be known as ‘Chinburg's Second Conjecture’ ([3], §4·2).

scholarly journals A non-abelian Stickelberger theorem

2010 ◽  
Vol 147 (1) ◽  
pp. 35-55 ◽  
Author(s):  
David Burns ◽  
Henri Johnston
Keyword(s):  
Galois Group ◽  
Group Ring ◽  
Galois Extension ◽  
Class Group ◽  
Number Fields ◽  
Stickelberger Theorem ◽  
Use Values

AbstractLet L/k be a finite Galois extension of number fields with Galois group G. For every odd prime p satisfying certain mild technical hypotheses, we use values of Artin L-functions to construct an element in the centre of the group ring ℤ(p)[G] that annihilates the p-part of the class group of L.

scholarly journals ERRATA: "GEOMETRIC GALOIS THEORY, NONLINEAR NUMBER FIELDS AND A GALOIS GROUP INTERPRETATION OF THE IDELE CLASS GROUP"

2011 ◽  
Vol 22 (02) ◽  
pp. 307-309
Author(s):  
T. M. GENDRON ◽  
A. VERJOVSKY
Keyword(s):  
Galois Group ◽  
Galois Theory ◽  
Class Group ◽  
Number Fields

In this errata we correct several mistakes (see Secs. 1 and 3–6 of this errata) that appear in the published version of our paper. The corrections have been implemented in the revised version [1]. In addition, in Sec. 2 we clarify an important point which was not adequately addressed in the published version; in Sec. 7 we point out an enhancement of the hyperbolization scheme included in [1]. The reader may also wish to consult [2].

scholarly journals ERRATA: "GEOMETRIC GALOIS THEORY, NONLINEAR NUMBER FIELDS AND A GALOIS GROUP INTERPRETATION OF THE IDELE CLASS GROUP"

2010 ◽  
Vol 21 (10) ◽  
pp. 1383-1385
Author(s):  
T. M. GENDRON ◽  
A. VERJOVSKY
Keyword(s):  
Galois Group ◽  
Galois Theory ◽  
Class Group ◽  
Number Fields

In this errata we correct several mistakes Secs. 1 and 3–6 that appear in the published version of our paper. The corrections have been implemented in the revised version [1]. In addition, in Sec. 2 we clarify an important point which was not adequately addressed in the published version; in Sec. 7 we point out an enhancement of the hyperbolization scheme included in [1]. The reader may also wish to consult [2].

scholarly journals The 2-rank of the class group of some real cyclic quartic number fields

2021 ◽  
Vol 131 (1) ◽  
Author(s):  
Abdelmalek Azizi ◽  
Mohammed Tamimi ◽  
Abdelkader Zekhnini
Keyword(s):  
Class Group ◽  
Number Fields

scholarly journals Explicit methods for the Hasse norm principle and applications to A n and S n extensions

2021 ◽  
pp. 1-41
Author(s):  
ANDRÉ MACEDO ◽  
RACHEL NEWTON
Keyword(s):  
Galois Group ◽  
Computational Methods ◽  
Number Fields ◽  
Normal Closure ◽  
Explicit Methods ◽  
Weak Approximation ◽  
Norm Principle ◽  
Theoretical Results

Abstract Let K/k be an extension of number fields. We describe theoretical results and computational methods for calculating the obstruction to the Hasse norm principle for K/k and the defect of weak approximation for the norm one torus \[R_{K/k}^1{\mathbb{G}_m}\] . We apply our techniques to give explicit and computable formulae for the obstruction to the Hasse norm principle and the defect of weak approximation when the normal closure of K/k has symmetric or alternating Galois group.

scholarly journals Phase Transitions on C*-Algebras from Actions of Congruence Monoids on Rings of Algebraic Integers

2020 ◽  
Author(s):  
Chris Bruce
Keyword(s):  
Phase Transitions ◽  
Class Group ◽  
Number Fields ◽  
Algebraic Integers ◽  
Ring Of Integers ◽  
C Algebras ◽  
Rings Of Algebraic Integers ◽  
Beta 2 ◽  
Factor State ◽  
Type Classification

Abstract We compute the KMS (equilibrium) states for the canonical time evolution on C*-algebras from actions of congruence monoids on rings of algebraic integers. We show that for each $\beta \in [1,2]$, there is a unique KMS$_\beta $ state, and we prove that it is a factor state of type III$_1$. There are phase transitions at $\beta =2$ and $\beta =\infty $ involving a quotient of a ray class group. Our computation of KMS and ground states generalizes the results of Cuntz, Deninger, and Laca for the full $ax+b$-semigroup over a ring of integers, and our type classification generalizes a result of Laca and Neshveyev in the case of the rational numbers and a result of Neshveyev in the case of arbitrary number fields.

scholarly journals On the $2$-class group of some number fields with large degree

2021 ◽  
pp. 13-26
Author(s):  
Mohamed Mahmoud Chems-Eddin ◽  
Abdelmalek Azizi ◽  
Abdelkader Zekhnini
Keyword(s):  
Class Group ◽  
Large Degree ◽  
Number Fields
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