Determination of Hauptmoduls and Construction of Abelian Extensions of Quadratic Number Fields

2007 ◽  
Vol 50 (3) ◽  
pp. 334-346
Author(s):  
Hung-Jen Chiang-Hsieh ◽  
Yifan Yang
Keyword(s):  
Number Fields ◽  
Cyclic Extension ◽  
Quadratic Number ◽  
Modular Functions ◽  
Genus Zero ◽  
Congruence Subgroups ◽  
Abelian Extensions ◽  
Quadratic Number Fields

AbstractWe obtain Hauptmoduls of genus zero congruence subgroups of the type (p) := Γ0(p) + wp, where p is a prime and wp is the Atkin–Lehner involution. We then use the Hauptmoduls, along with modular functions on Γ1(p) to construct families of cyclic extensions of quadratic number fields. Further examples of cyclic extension of bi-quadratic and tri-quadratic number fields are also given.

scholarly journals Hyperbolic tessellations and generators of for imaginary quadratic fields

2021 ◽  
Vol 9 ◽  
Author(s):  
David Burns ◽  
Rob de Jeu ◽  
Herbert Gangl ◽  
Alexander D. Rahm ◽  
Dan Yasaki
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Quadratic Fields ◽  
Quadratic Number ◽  
Imaginary Quadratic Fields ◽  
Quadratic Number Fields ◽  
Formula Group ◽  
Imaginary Quadratic Number ◽  
Direct Calculations

Abstract We develop methods for constructing explicit generators, modulo torsion, of the $K_3$ -groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic $3$ -space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite $K_3$ -group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for $ K_3 $ of any field, predict the precise power of $2$ that should occur in the Lichtenbaum conjecture at $ -1 $ and prove that this prediction is valid for all abelian number fields.

scholarly journals On quaternion algebras over the composite of quadratic number fields

2021 ◽  
Vol 56 (1) ◽  
pp. 63-78
Author(s):  
Vincenzo Acciaro ◽  
◽  
Diana Savin ◽  
Mohammed Taous ◽  
Abdelkader Zekhnini ◽  
...  
Keyword(s):  
Number Fields ◽  
Complete Characterization ◽  
Quadratic Number ◽  
Quaternion Algebras ◽  
Quadratic Number Fields

Let p and q be two positive prime integers. In this paper we obtain a complete characterization of division quaternion algebras HK(p, q) over the composite K of n quadratic number fields.

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