New analytic problems over imaginary quadratic number fields

Number Theory ◽  
2013 ◽  
Author(s):  
Yoichi Motohashi
Keyword(s):  
Number Fields ◽  
Quadratic Number ◽  
Quadratic Number Fields ◽  
Imaginary Quadratic Number

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.

Nonadjacent Radix-τ Expansions of Integers in Euclidean Imaginary Quadratic Number Fields

2008 ◽  
Vol 60 (6) ◽  
pp. 1267-1282 ◽  
Author(s):  
Ian F. Blake ◽  
V. Kumar Murty ◽  
Guangwu Xu
Keyword(s):  
Number Fields ◽  
Quadratic Number ◽  
Efficient Point ◽  
Koblitz Curves ◽  
Quadratic Number Fields ◽  
Imaginary Quadratic Number

AbstractIn his seminal papers, Koblitz proposed curves for cryptographic use. For fast operations on these curves, these papers also initiated a study of the radix-τ expansion of integers in the number fields and . The (window) nonadjacent form of τ -expansion of integers in was first investigated by Solinas. For integers in , the nonadjacent form and the window nonadjacent form of the τ -expansion were studied. These are used for efficient point multiplications on Koblitz curves. In this paper, we complete the picture by producing the (window) nonadjacent radix-τ expansions for integers in all Euclidean imaginary quadratic number fields.

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