scholarly journals An explicit family of cubic number fields with large 2-rank of the class group

2018 ◽  
Vol 182 (2) ◽  
pp. 117-132 ◽  
Author(s):  
Avinash Kulkarni
Keyword(s):  
Class Group ◽  
Number Fields ◽  
Cubic 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.

scholarly journals Class-group problems for cubic number fields

1997 ◽  
Vol 23 (2) ◽  
pp. 365-378
Author(s):  
Stéphane LOUBOUTIN
Keyword(s):  
Class Group ◽  
Number Fields ◽  
Cubic Number Fields

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

The densities for 3-ranks of tame kernels of cyclic cubic number fields

2013 ◽  
Vol 57 (1) ◽  
pp. 43-47 ◽  
Author(s):  
XiaoYun Cheng ◽  
XueJun Guo ◽  
HouRong Qin
Keyword(s):  
Number Fields ◽  
Tame Kernels ◽  
Cubic Number Fields

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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