scholarly journals The Hecke Group H λ 4 Acting on Imaginary Quadratic Number Fields

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Abdulaziz Deajim
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Relevant Literature ◽  
Number Fields ◽  
Quadratic Number ◽  
Hecke Group ◽  
Quadratic Number Fields ◽  
Imaginary Quadratic Number ◽  
Imaginary Number

Let H λ 4 be the Hecke group x , y : x 2 = y 4 = 1 and, for a square-free positive integer n , consider the subset ℚ ∗ − n = a + − n / c | a , b = a 2 + n / c ∈ ℤ ,   c ∈ 2 ℤ of the quadratic imaginary number field ℚ − n . Following a line of research in the relevant literature, we study the properties of the action of H λ 4 on ℚ ∗ − n . In particular, we calculate the number of orbits arising from this action for every such n . Some illustrative examples 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.

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.

scholarly journals Geometric-progression-free sets over quadratic number fields

2017 ◽  
Vol 147 (2) ◽  
pp. 245-262
Author(s):  
Andrew Best ◽  
Karen Huan ◽  
Nathan McNew ◽  
Steven J. Miller ◽  
Jasmine Powell ◽  
...  
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Geometric Progression ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Ring Of Integers ◽  
Quadratic Number Fields ◽  
Geometric Progressions ◽  
Upper Density ◽  
Free Sets

In Ramsey theory one wishes to know how large a collection of objects can be while avoiding a particular substructure. A problem of recent interest has been to study how large subsets of the natural numbers can be while avoiding three-term geometric progressions. Building on recent progress on this problem, we consider the analogous problem over quadratic number fields. We first construct high-density subsets of the algebraic integers of an imaginary quadratic number field that avoid three-term geometric progressions. When unique factorization fails, or over a real quadratic number field, we instead look at subsets of ideals of the ring of integers. Our approach here is to construct sets ‘greedily’, a generalization of the greedy set of rational integers considered by Rankin. We then describe the densities of these sets in terms of values of the Dedekind zeta function. Next, we consider geometric-progression-free sets with large upper density. We generalize an argument by Riddell to obtain upper bounds for the upper density of geometric-progression-free subsets, and construct sets avoiding geometric progressions with high upper density to obtain lower bounds for the supremum of the upper density of all such subsets. Both arguments depend critically on the elements with small norm in the ring of integers.

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